SUME ENTERRELATIONS 0F
FERTILIZERS. CARBOHYDRATE
PRODUCTION AND YIELD OF

FIELD BEANS

THESIS FOR THE DEGREE OF M. s.
C. W. Lauritzen
1932

   

‘(Lv‘qgguvmdé €37.

MIRA!" A.“ an mum at.

 

 

SOME INTERRELATIONS OF FERTILIZERS,
CARBOHYDRATE PRODUCTION AND YIELD

OF FIELD BEANS

by

.1
y .
‘\

C'.‘ w.’ LAURITZEN
rTmL‘“

A THESIS
PRESENTED TO THE FACULTY
OF THE
MICHIGAN STATE comer:
OF
AGRICULTURE AND APPLIED SCIENCE
IN PARTIAL memmm
or THE REQUImmMS
FOR THE

DEGREE OF MASTER OF SCIENCE

East Lansing

1 9 3 2

THFQ‘S

 

ACKNOWLEDGMENT
The writer wishes to express his appreciation to
Dr. C. H. Spurway for his invaluable guidance and interest
shown during the course of this investigations Thanks is
due to Dr. C. E. Millar and Dr. J. W. Grist for the critical
reading of the manuscript, also to many others who have
contributed to this research by their suggestions and criti-

cisms.

0 (j ‘ “a;
-r .
1 N __ l I

-1-
IHTRODUCTIOK

The yield obtained from any economic plant is dependent upon
the many factors which influence the growth of the crop under con-
sideration. Commercial fertilizer applications may be one of these
factors, depending on whether there exists a soil deficiency of the
elements supplied. If applications of a commercial fertilizer
produce a marked effect upon the yield, the other factors remaining
unchanged, it is assumed that the plant nutrient elements supplied
have or have not been limiting factors in the production of the
crop, depending on the yield changes produced by the fertilizer.

Carbohydrate production is undoubtedly a necessary process in
the growth of plants. Since fertilizer elements affect plant
growth the question is raised as to what part these fertilizer
elements play in the synthesis of the various carbohydrates found
in plants and what relation exists between carbohydrate production
in growing plants and the yields obtained. In order to determine
if such relationships exist in the case of bean plants, and also
to determine if the rate of production of carbohydrates at dif—
ferent times of the day is influenced by the fertilizer elements
supplied, bean plant parts were analyzed for carbohydrates, nitrogen,
'and dny weight in plant samples taken at four-hour intervals from
check plots and from plots receiving applications of various
fertilizer elements.

.A considerable amount of research on carbohydrate production

in plants has been carried out using several kinds of plants.

-2-

These researches have been.mainly attempts to determine the
order of synthesis of the different plant carbohydrates and
certain relations they bear to one another; as the variation
in the amounts present in the plants with the change of tempera-
ture, intensity of sunlight, and consequently the time of day.
Other investigations made along this line have had for their
purpose the determination of the functions of nutrient elements
in the synthesis of carbohydrates in plants.

No one, to our knowledge, has attempted to correlate carbo-
hydrate production during periods of growth with.the eventual
yield; or has used field beans in researches dealing with carbo-
hydrate production. Beans are a valuable food crop high in
carbohydrate content, and they are extremely sensitive to environ-
mental factors which cause uncertainty in their profitable pro-
duction. Because of the importance of this crop from the stand-
point of both production and consumption, studies of this nature

are desirable.

-3-
HISTORICAL

Noble (9), working with crops that store large amounts
of carbohydrates such as sugar beets and mangolds, found
that a sufficient supply of potassium was necessary for the
normal production of carbohydrates in the plants.

Stocklasa (16) showed that green plants did not have
the power of synthesis in the absence of potassium.

Reed (11) found evidence in the plants he studied that
potassium was necessary for the formation of starch, but gave
no evidence that phosphorus was necessary for starch.formation.
He showed, however, that a deficiency of phosphorus seriously
interfered with the transformation of starch into water-
soluble carbohydrates.

Weiszmann (18) made the same observation as Stocklasa.

Smith and Butler (1h) could not find evidence to support
Stockflasa and Weiszmann.

Schuster (12) pointed out that the results obtained from
an application of potassium to a crap depended on the amount
of available potassium in the soil as well as other soil
properties, and stressed the importance of considering the soil
type as a significant factor in planning an investigation of
this nature.

Janssen and Bartholomew (h) found that tomato plants grown
in mediums low in potassium.showed an increase in the per cent

of dry matter, nitrogen, and sugars. They also reported (5)

-h-

that the percentages of carbohydrates and potassium in the
plants fluctuated greathy and concluded that a.high.per cent
of sugars and starch was not necessarily associated with a
high per cent of potassium.in the plant. They found however
a fairly good correlation between the total weight of sugars
and starch per plant and the potassium content.

MacGillivary (6) showed that in the case of tomatoes a
deficiency of phosphorus resulted in an increase in the per
cent of total nitrogen, this increase being mostly water soluble
forms of nitrogen, excluding nitrates, but caused a decided
increase in the per cent of sugars.

Stark (15) noted that when potassium was applied to soy-
beans, in addition to limestone and organic matter, a decrease
in the per cent of protein resulted; but limestone and organic
matter alone gave an increase in the per cent of protein.

Davis, Daish, and Sawyer (2), after making a study of the
formation and translocation of sugars in mangolds, reported that
the proportion of saccharose in the leaf tissue followed the
temperature curve closely during the daytime; and that the pro-
portion of hexoses increased faster than the temperature, and in
such a.manner that the curve showing the ratio of hexoses to
saccharose in practically parallel to the temperature curve.

Davis and Sawyer (3) found in their study of potato plants

that saccharose was the principal sugar in the plant when the

-5-

tubers were being formed, and that the amount increased from
sunrise until 2 P.M. following approximately the rise in tempera-
ture. The starch content of the plants reached its maximum point
two hours before sunset. From that time on the amount of starch
decreased till there was very little present at 2 A.M.

{ason and Maskell (7) studied the transport of carbohydrates
in the cotton plant. They found a significant lag in the diurnal
variations in the bark over those in the leaves and that the ‘
variation of the total sugars was due largely to the variation in
the concentration of sucrose.

The rate of transport of carbohydrates from the bark to the
boll in cotton plants was found to correlate with the variation
in the sucrose gradient from the bark to the boll, the rate being
about four times as great in the day as in the night.

Clements (1), working with potato, soybean, and sunflower
plants, showed by his graphs that the period of maximum carbohydrate
production was from noon until about 9 P.M. The increase in starch
content was especially noticeable during this period. A distinct
correlation was found between favorable environmental conditions
and the formation and translocation of carbohydrates in the plant.

Cole (19) very recently working with the potato plant, found
irregular variation in the sugar content of plants grown on plots
receiving various fertilizer treatments but no differences were
found which could be attributed to fertilizer application. He
expresses the Opinion that under field conditions other factors

such as intensity of light, moisture, and humitity enter in to

-6-

cause variations in the sugar content of the potato plant to
a greater extent than does the relative supply of nutrient elements

present.

1 rd.. 3.12511 - 23618.. - .

- 2.1200,- -‘

 

 

 

 

 

 

 

 

 

 

FIGURE I
1010 101A 1013
o - 16 . 0 POUO
1020 102‘ 1023
o - 16 ' 0 3.0.
fl
43?
4’ 1030 103A 1033
§
{‘3 0 - 16 - 8 P.U.
.8
O
3 101m 10111 10113
03
«‘3
a O " 16 b 8 3000
,o
3
A 1050 1051 1053
O
E
a
a 0 - 16 - 8 P.U.
8 h - o 4- 0 3.00
M
g 1060 1061 1063
3
m
M - 16 «3 PM.
1070 107A 1073
I" - 16 "' 8 30°.
KEY 1030 1031 10813
P.U. Plowed under
3.0. Broadcast 1" - 16 u 8 Pen.
1090 1091 1093
)4 - 16 ‘L 8 3.6.

 

 

 

 

 

 

v/ °sqtoou

 

1/4111 oo

-7-

EXPERIMENTAL PROCEDURE

The bean plant samples used in this research and the
yield results were obtained from c00perative fertility plots
located in the vicinity of Ithaca, Michigan, on the soil type
known as Brockston loam. Figure 1 shows the plan of the
fertility field, the kinds and amounts of fertilizers applied,
the methods of applying the fertilizers, and the location of
the check and treated plots from.which the plant samples were
taken for analysis and from which the yields were obtained.

Judging from the yields of crops produced previously on
this land, the field was above the general average in fertility
for the locality. The bean yields reported in this thesis,
however, are much.be10w the average for the Brookston soil type
in this State, due probably to the deficiency of rainfall during
the growing season of this year (1930). During the year pre-
ceding the establishment of the bean fertility plots this field
was cr0pped to corn and was plowed early in the Spring of 1930
for beans.

The bean fertility plots were laid out and the fertilizers
applied in such a way as to make possible comparisons of yields,
and carbohydrate production of beans on soil receiving no treat-
ment; superphosphate alone; superphosphate and.muriate of potash;
and nitrate of soda, superphosphate, and muriate of potash.
Three methods of fertilizer application were used; namely,

(1) broadcast before plowing, (2) broadcast on the plowed and

-g-

worked field before seeding, (3) superphosphate and muriate
of potash plowed under and nitrate of soda broadcast on the
plowed and worked field before seeding. ,All treatments were
applied at the rate of four hundred pounds per acre, except
on plots 108 (A.and B) and 109 (A and B), which were ferti-
lized at the rate of two hundred pounds per acre. The ferti-
lizers were mixed according to the following formulas: 0-16-0,
0-16-8, and M-lG-S. Then they were carefully broadcast by
hand on the plots, having first been diluted by mixing them
thoroughly with about twice their volumes of moist soil to
prevent any possible blowing and to insure a more uniform
distribution over the plots. The beans were planted June 12,
1930, and harvested September 23, 1930.

The plots used to obtain plants for carbohydrate analysis
were 102A, lOHA, 107A, and the check strip as designated (Figure
1). Plots 1023, thB, and 1073 were harvested and the yields
determined for comparison with the carbohydrate production of
the plants on the.A portion of the corresponding plots. The
yields on plots 1010 to 1070 inclusive, were averaged and this
average value taken for comparison with the carbohydrate pro-
duction on the check strip.

The day preceding sampling was warm.and.part1y cloudy.
During the period of sampling the weather was clear and warm,
except for the period between four o'clock P.M. and eight

o'clock P.M., when it was partly cloudy;

-9-

The bean plants at the time of sampling were green
having pods somewhat more mature than string beans when taken
for table use. Some variation was observed in the maturity
of the plants as is always the case; but on the average, it
might be said that the plants were of such maturity that the
pods contained beans from one-half to full size. The plant
sglples were taken every four hours during a period of twenty-
four hours. The first samples were taken at midnight August
13, 1930, and the last samples at midnight August 1h, 1930;
thus, the first and last samples were taken at the same time
on two different days. The samples were collected by taking
plants promiscuously over the plots, the several samples being
gathered at one time and taken as quickly as possible to the
field laboratory close by where the equipment for preserving
the samples was located.

The leaves, stems, and pods of the plant were separated.by
hand. Fifty grams of each part which.previously had been cut
into pieces one to two inches in length were placed in pint
Jars containing approximately one gram of Ca003 to neutralize any
acids present which would tend to hydrolyze the sugars; the
presence of an excess of 0a003 being unimportant as its dissociation
was such that the pH was not raised enough to cause the breaking
down of the sugar molecule which occurs at higher concentrations
of hydroxyl ions. The Jars were then nearly filled with boiling

95% ethyl alcohol; the caps turned on loosely and the Jars set in

--10--

a water bath hot enough to keep the alcohol at the boiling
point in order to prevent all enzymic action and partially
extract the sugars from the plant tissues. At the end of one
hour the jars were removed from the water bath and enough more
hot alcohol added to cover the plant tissue, in case some had
been lost due to evaporation. The Jar caps were then tightened

down and the jars placed in storage until wanted for analysis.

-11-
ANALYTICAL METHODS
Analysis of Bean Tissue

Preliminary treatment of the sample:

The field sample preserved in alcohol was filtered into
a 1000 cc. volumetric flaSk and.washed with 80% alcohol until
the filtrate was about 950 cc. This volume was made up to
1000 cc. and was called the ”extract”.

The residue was placed in a 200 cc. weighed beaker and
dried in an oven at 65° 0. for M8 hours, cooled in a desicator
and weighed. The aliquots for analysis were calculated on the
basis of this weight.

Treatment of the Extract

The alcohol from 300 cc. of the alcoholic extract was
distilled off under reduced pressure in a classion flask below
65 0. When about 50 cc. remained the residue was washed from
the classion flask into a 100 cc. volumetric flask, with warm
water, cooled, made up to volume and transferred to a 100 cc.
centrifuge tube. About 0.2 grams of anhydrous lead subacetate
(Horn's dry lead) was added, mixed well, and the samples centri-
fuged for five minutes. The clear liquid was transferred to
another centrifuge tube and de-leading by adding about 0.5 grams
of sodium oxalate, mixing and centrifuging five minutes. The
clear solution was decanted off into a beaker and 75 cc. of it

taken and placed in a 200 cc. volumetric flash; ‘Water was added

-12..

to make the contents of the flask up to volume.
The Determination of Dextrose

Fifty cc. of the solution were placed in a 300 cc.
Erlenmeyer flask and 50 cc. of Fehling's solution were
added—-25 cc. of EA” (69.28 grams CuSQu.5déO dissolved in
water and diluted to 1 liter) and 25 cc. of ”B" (3M6 grams
Rochelle salts and 100 grams NaOH dissolved in water and
diluted to 1 liter). The flask was placed in a water bath
at 80°C. for 30 minutes (10). At the end of this time the
flasks were removed and cooled to about #000.

The determination was continued using the Schaffer and
Hartmann.method (l3). Twenty-five cc. of an iodide-iodate
solution (60 grams KI, 5.2 grams K103, and a few drops of
concentrated NaOH dissolved and diluted to 1 liter with.water)
was added with shaking. Next, 15 cc. of 8N HéSOu were added
and the flask shaken until the Cu20 was dissolved. .Then 20
cc. of a saturated solution of K2020h were added and the
contents of the flask titrated at once with standard‘ NaZSZO3 sol-
ution (2M.82 grams NaZSZOjfifléo dissolved in water, a few drops
of concentrated NaOH added and the volume made to 1000 cc. with
water).

The burette reading was the number of cc. of 178128203 used to

oxidize the excess of iodine present which was the amount liber-

.- Jour. Assoc. Official Agri. Chem. 12: 169 (1929).

-13-
ated by the acid less the amount used in oxidizing the Cu20.
The amount of iodine which reacted with the Cu20 was obtained
by subtracting the number of co. Na25203 taken to titrate the
sample from the number of cc. Na25203 taken to titrate the
blank determination. The blank consisted of 50 cc. of water
run in duplicate, with each set of determinations which re-
ceived exactly the same treatment as the samples and gave the
total amount of iodine liberated.
0112501,} 21: 01112
211825203; 1: NaQSuO6I- 2Na 1

Since one equivalent of Cu required one equivalent of I
for oxidation, the difference in the titration of the blank and
the sample in cc. multiplied by the Cu equivalent in.milligrams
per cc. of the NagSng solution equalled the Cu reduced by sugars
and was converted to dextrose by referring to the lunson and
Walker tables (8). The dextrose in the determination times
17.777 (dilution factor) equals dextrose in total sample.

When the titration is referred to in cc. of Na25203 it
should be understood that it is the co. difference in the titrae
tion of the sample and its corresponding blank determination.

The Determination of Total Sugars

Fifty cc. of the same solution used for the determination of
dextrose in the extract was placed in a 300 cc. Erlenmeyer flask.
Five grams of citric acid crystals were added and the contents of
the flask boiled quietly for ten.minutes, then cooled and neutral-

ized with neon using phenolphtalein as an indicator. The deter-

-1h-

mination is continued as for dextrose except 20 Cc. of 8N
sésoh instead of 15 cc. are used. The cc. Na28203 times Cu
equivalent of Na25203 solution equals Cu reduced in the deter-
mination. Dextrose equivalent of the Cu times 17.777 (dilution -
factor) equals total sugar in sample in terms of dextrose. The
total sugar minus the dextrose times 0.95 equals hydrolyzable
sugars or sucrose.
The Determination of Dextrine

The dry residue was ground in a Wiley mill and a three-
tenths aliquot weighed out in a 150 cc. beaker, 75 cc. of water
and a few cc. of toluene were added and the sample allowed to
stand over night. The solution was filtered off into a 250 cc.
volumetric flask and the residue washed with warm water until
the flask contained about 150 cc. of the filtrate. To this
filtrate was added sufficient concentrated HéSOh to make it 25%
of HéSOu. The mixture was shaken and let stand over night, then
made up to volume and a 50 cc. aliquot withdrawn and neutralized
with concentrated NaOH using phenolphthalein as an indicator.

The contents were allowed to cool and the determination for
dextrose made at once. The number of co. Na25203 times Cu
equivalent of Na25203 solution equals Cu in determination. The
dextrose equivalent timOI 16.666 (dilution factor) equals dextrine
in sample reported as dextrose.

The Determination of Starch

The residue from the dextrine determination was transferred

to the original 150 cc. beaker using about 50 cc. of water and

-15-
the starch in it determined by the Take-Diastase Method (17).

In this method the beaker containing the residue was heated for
#0 minutes on a steam bath with stirring. After cooling, enough
TakaeDiastase to digest the starch (0.1 gram Taka-Diastase to

0.3 grams of starch) and 3 cc. of toluene were added. The
samples were incubated in a constant temperature oven at 3830 for
an hours, with stirring at intervals, more toluene being added

to replace that lost by evaporation. After digestion was com»
pleted the mixture was heated to boiling on a steam bath and
boiled for 15 minutes; then the residue was filtered into a 250
cc. volumetric flask and washed until nearly that volume is ob-
tained. Water was added to complete the volume and a 50 cc. ali-
quot taken for the determination of dextrose.

The total starch in 50 cc. equals 51 f 52(17)

2. 31 .

E? Total reducing power of the digested starch.

The starch in 50 cc. is multiplied by 16.666 (dilution
factor) to obtain starch in original sample of bean tissue.

Determination of Dry Weight

100 cc. of the extract representing a one-tenth aliquot
was placed in a weighed 200 cc. beaker (tall form) and the
alcohol evaporated off at a low heat on an electric hot plate.
A.one-tenth aliquot of the corresponding dry ground tissue was

then added to the same Beaker and the contents of the beaker

-16-
dried at 105°C. for 2% hours, cooled in a desiccator and
weighed. From.these weighings the per cent dry weights were
calculated.
Determination of NitrOgen

The loose, dry residue from the dry weight determination
above was transferred from the beaker to an 800 cc. Kjeldahl
flask to which was added 15-18 grams of KZSOM and 1 gram of
Cusou. The residue dried on the beaker was washed off by first
warming with 15 cc. of concentrated HZSON and then dislodged by
means of a rubber policeman and another 15 cc. of cold concen-
trated Hésou and finally washed out with a few cc. of water.
From this point the procedure was the same as that employed in

the determination of nitrogen by the Kjeldahl method.

DISCUSSION

The results of this research showed some evidence of
a relation between the percentages of sugars, starch, and dry
weight in the bean plant parts and the time of day. .Also, that
the average amount of starch found in the plant parts was related
to the fertilizer treatment and the yield of beans.

Contrary to the results obtained in 1915 by Davis, Daish,
and Sawyer (2), who worked with.mangolds, it will be seen by
observing graphs 1 and N that the per cent of sugars in the bean
leaves at different times of the day did not follow the daily
temperature changes, but, if anything, tended to be low during
periods of high temperature and high during periods of low tempera~
ture. If the stems and pods alone were considered, there was a
noticeably low content of sugars during the period from 8 A,r. to
8 9.1.1. (graphs 2, 3, 5, 6, 19, 20). In their work with potatoes,
Davis and Sawyer (3) obtained similar results for sugars in the
leaves of mangolds and reported in addition a general increase
in the per cent of starch from sunrise until two hours before
sunset. This work on beans showed some indication of a maximum
starch content in the leaves at about 8 P.M. (graphs 10 and Ml);
but a period of low starch content was shown during the middle of
the day.

The stems and pods of the beans did not show a maximum
starch content as in the case of leaves at 8 P.M., but rather at

some time between 12 K. and N Alh., and a minimum point at 12fi.

-13-

(graphs 10, ll, 12, 22, ”1, and MB). The variations in per cent
dry weight followed the temperature changes although with some
degree of lag; consequently it seemed that a higher per cent of
water in the plants was associated with a higher per cent of
sugar and starch. During the entire sampling period of 2% hours,
the per cent of dextrine and nitrogen was fairly constant in all
the plant parts and no variation due to treatments was evident.

It will be noticed that a collective average of the per cent
constituents in the leaves.stems, and pods was made. However,
it was known that this average did not represent the exact per
cent of the constituents in the whole plant, but was taken for
the purpose of comparison, as it tended to produce uniformity in
the results and conclusions, thus causing the curves of the per
cent constituents with the time of day to become smoother and
the results of the several treatments to more nearly parallel each
other.

Considering the average per cent of the several constituents
determined in leaves, stems, and pods separately, some very notice-
able relationships become evident, as to the relative amounts of the
constituents in the different plant parts. Dextrose was found to
be present in the greatest amounts in the stems and the least
amounts in the leaves (graphs 1, 2, 3, 25, and 37). Sucrose, on
the other hand, although contained to the least amount in the
leaves, existed in greater amounts in the pods than in the stems

(graphs h, 5, 6, 26, and 38). The greatest per cent of starch was

-19-

in the pods with the least amount in the stems. Carbohydrates,
which were the sum of the constituents named above and dextrine,
which was present in nearly the same amount in each.plant part,
increased in amounts from leaves to stems to pods. The per

cent dry weight followed the same order in the plant parts as
did the per cent carbohydrates. Thus, it will be seen that the
bean plant tissue which contained the least water had the largest
per cent of carbohydrates, and yet, as was previously shown, the
water and carbohydrate content of the plant parts fluctuated
tagether. Nitrogen was present in about the same percentages

in the leaves and pods with considerably less in the stems.

The above relationships were discovered from data gathered
primarily to determine the effects of fertilizers on the per
cent of certain plant constituents at different times of the
day, but there seemed to be no apparent relationship shown in
this respect, and no attempt will be made here to explain these
data on the basis of plant processes.

The fertilizer treatments did not appear to have any distin-
guishable effect on the per cent sugars in the plant except perhaps
the 0-16-8 (graph 25). In this case there was evidence that the
per cent dextrose was depressed. When the daily average was con-
sidered the per cent starch in the bean plant parts showed a
rather definite correlation with the fertilizer application. The
bean plants grown on the plot receiving an application of super-

phosphate and muriate of potash tOgether showed an increase in

-20=

the per cent of starch over those grown on the plots receiving
superphosphate alone and those grown on the plots receiving no
treatment (graphs 29, 35, MN, M5). The addition of nitrate of
soda to superphosphate and muriate of potash further increased
the per cent of starch in the leaves and stems, but depressed
the per cent in the pods. The depressing effect of nitrogen
in the fertilizer treatment on the per cent starch in the bean
pods might be accounted for in that the plants on this plot may
have been a little less mature and starch was not being accumu-
lated there to the extent it would have been if the plants had
been more mature. The plants on this plot seemed a little
greener at the time the samples were taken than the plants on

the other plots.
Considering the fertilizer nutrient elements separately, po-

tassium in the fertilizer apparently increased the per cent of
starch in the bean plant parts (11); or at least a combination

of phosphorus and potassium increased the per cent starch over

no treatment or phosphorus alone. The increase in the yield of
beans paralleled the increase in the per cent starch and carbo-
hydrates, as a whole, in the bean plant parts (graphs uh, M5,

and #6) (tables u, 7, and 8). Therefore, the yield seemed to be
related to the per cent starch and also to the total carbohydrates
in the bean plant parts at this stage of maturity. Both the per
cent carbohydrates at this time, and the yield, were related to

the fertilizers applied.

-21-

SUMMARY

A study was made of carbohydrate production, per cent
nitrOgen, and the dry weight of the bean plant relative to
the time of day and fertilizer application. A.relation
was found between carbohydrate production and the time of
day, but no apparent relation was evident between fertilizer
application and percentage constituents at various times of
the day. It was found, however, that there was a marked
difference in the per cent composition in the leaves, stems,
and pods of the plant, and that the relation varies with
the constituent under consideration.

It appeared that potassium did increase the percentage
of starch and total carbohydrates in the bean.plant parts
and that this increase was reinforced by the addition of
nitrogen or perhaps it should be said that phosphorus and
potassium together gave an increase in carbohydrates which
was further increased by the addition of nitrogen. This
increase in carbohydrate production at this period of growth

was paralleled by a corresponding increase in yield.

Per cent dextrose in bean plant parts from check and
fertilized field plots calculated on the dry weight

basil 0

TABLEI

 

Average of
Treatments...

1L--16--8

0-16-0 0-16-s

Check

 

Time

Leaves

 

 

70272715
88868638
0 O O O O o O 0
00000010
5 3 1.1
375m9®29
0 0.00 O. 0
00100010
7.38 812‘...
(wow/aw SJl-l.

0.00.0.0.0.1.0

 

1.01.
0.95
0.69
0.87
0 81
0.66
1.1.1
0.92

 

mmmfi gap...)

.0....90
1.0000010

 

1+ 1.21.
8 ‘0“.
12 n.

1t PM.
8 P.M.
12 14.
Average

12!.

Stems

lenne 31.66
2 8 lrhw 52 8

31.122222fl2

 

73 368k.
mégmuaboamfll

1.12.12 95.22 2 31.01

 

578
.47mwh. 7h. .nw

329..1.?..22../2

 

Mm. mwm %r%ur=w%

333.1. 9... 2 3?.

 

21. 3005
1.8.1 9.1.27.

33222232

 

12 M.

u 1.11.
8 1.3.
12 N.

h P.M.
8 P.M.
12 u.
Average

Pods

mum/kn 925%

1211.10.11.

 

easieeao

o.21.1.1.0.1.1.

 

37.658 756
391. 378 32

11110.0.11.

 

 

13310931
07382976
0 O. O 000
21101021
.40 936
8.43

122?.

o 0
221011

 

1. 1.1.4.
8 1.16.
12 N.

14 P.M.
8 POM.
12 1.1.
Average

12M.

Average of
Treatmen

 

17-16-8

 

 

Leaves
Stems

0-16-0 0-16-8

 

fertilized field plots calculated on.the dry weight
Check

basis.

 

TABLE 2 Per cent sucrose in bean plant parts from check and

 

 

Time

 

 

 

 

 

 

 

 

 

 

90672725 Ion-8371.71.
6710115814. 36352530
0 C O C O O C O . O C .....
22222222 33322233
62989751. 29
antennae awn 27
22221332 231W2121..J2
1.615 1.73 1135567
2637 .6su edeeeee.
3221.1«29HZ 8 33321233
d
O
P
02 909 623 “1.
.usss 5s5.JJmm
2&122122 “331.2133
79 61. ”95055
2.7. 8M....4..%. 7.9nw81x4.%1
2221.29H9m?“ 9H3zfl239m33
mm .. m
. .uu . . . . .
u..n..ur . . .m
A... PD. 0 MNJNvmnrnr

Per cent dextrine in bean plant parts from check and
fertilized field plots calculated on the dry weight

basis .

TABLE 3

 

Average of
Iregtmegts

4-16-8

0-16-8

Check 0-16-0

 

Time

Leaves

 

 

 

 

1.. 73516 56
6 96 8 7006 7

........
00000000

 

1. w.
8 1.1!.
12 N.

1‘ PO”.
8 PJI.
12 M.
Average

12”.

Stems

 

........
00011110

 

M1157 39
.../580.19 90w

.. ...
01000000

 

 

7.68.466 3
.4 40738 In». W.

. . . . .
01000111

 

8 P.M.

12 M .
Average

4.A.M.
s 1.11.
12 N.

1. 9.3.

1211.

Pods

$95738
.190119 .

. . . . . .
01011101

 

 

 

 

 

TABLE h Per cent starch in bean.plant parts from.check and
fertilized field plots calculated on the dry weight

 

 

b33130 . '
Time Check 0-16-0 0-16-8 11-16-8 gm
Leave:
12M. 0.80 0.97 Luna 11.61 2.70
h 1.11. .99 11.80 5.36 11.90 9.76
8 1.11. .78 n.61 6.08 5.92 2.22
12 N. 11.61 5.02 3.37 3.33 .08
u 2.24. 5.3 3.79 2. 7 3.87 3.88
8 2.14. 7.8 3.69 5.30 11.80 7.16
12 u. 1.15 1.05 6.59 “($8 11.07
Ayerage ”.06 3.N2 n.80 5.92 ”.55
Stems
12 u. 0.79 1.63 14.314 11.10 2.72
h 1.11. 14.56 5.3 2.27 5.16 11.33
8 1.8. 11.76 3.02 11.32 3.25 3.91
12 n. 2.70 3.22 2.96 2. 2 2.82
n P0“. 3.8” 3060 2.10 ' 3093 3037
8 1m. .10 3.142 11.26 .78 .111
12 M. 3.22 1.56 6.02 7.97 14.69
Average 3.h2 3.11 3.75 n.56 3.71
Pods
1211. 11.95 9.81& 9.78 8.73 8.32
n 1.11. 9.52 9.95 9.01 8.69 9.29
8 1.14. 5.99 7.77 7.99 6.86 7.03
12 N. 5.60 11.97 6.87 11.117 5.118
b. 2.11. 5.61 H.87 6.90 6.12 5.88
8 P.M. 7.10 5.21» 9.62 8.79 7.69
12 M. 10.09 11.03 12.16 12.70 11.19
Average 6.98 7.67 8.83 8.05 7.88

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

TABLE 5 Per cent dry matter in been plant parts from cheek
and fertilized field plots.

 

 

nme Check 0-16-0 0-16-8 11.168 f”? ° 0| 1'
Leaves
12 M. 19.80 20.23 19.911 19.90 19.l+7
n 1.11. 19.69 19.99 18.61 20.50 19. 69
8 1.11. 20.86 19.59 18. 111+ 18.78 19.112
12 N. 21.07 20.97 21. 3’4 20. 78 21. 01+
11 2.11. 20.25 19 .911 21.16 20. 60 20.149
8 19.11. 20.10 19.13 20. 99 21.71 20.118
12 M. 19.50 19.06 21.38 21.02 20.211
1verege 20.18 19.811 20.27 20.117 20.19
Stems
12 m. 22.28 21.55 22.37 22.0 22.06
n 1.11. 21.111 21.62 20.96 22.1 21.53
8 1.11. 23. 51 20.91 22.31 20.711 , 21.87
12 n. 2.29 22.96 23.16 23.06 23.12
n ma. .09 23.611 .26 23. 67 23.67
8 17.21. 21.21 21.97 2 .91 22.111 23. 8
12 11.1.22 19.60 22.08 22.60 21. 6
Average 22. 21.75 22.72 22.38 22.33
Pods
12 M. 26.15 25.98 27.08 28.57 26.95
1 1.8. 21.93 25.21 23.76 211.25 34.31
8 1.11. 25.17 26.93 2 .113 25.03 25.39
12 n. .25 79 29.09 28.36 27.143 27.67
11 2.11. 25.145 26.140 28. 50 26.88 27.31
8 2.11. 23.1 29.911 27. 59 30.112 27.77
12 n. 27.7 25.55 28.16 27.53 27.25
1verage 25.05 27.31 27. 12 27.16 26. 66

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

fertilized field plots calculated on the dry weight

TABLE 6 Per cent nitrogen in bean plant parts from.check and
basis.

 

Ayerage of

Tr

Check 0-16-0 0-16-8 4-16-8

Time

 

Leaves

 

 

3333332 3

 

666 76 78 1..
#6 J?— 350/1...“

1221212222222222

 

wwwmm $1“..an

O O
IJsz/zazzzzzzzz

 

4 1.11.
8 1.11.
12 NO

4 2.11.
8 PM.
12 u.
Awerags

12 M.

Stems

367101.. 300
1001.9 oM/OJ

22221211..

 

9 9511.6
nu. 9J1hfl8

11.21.1211

 

71757.66
992 0.0. 2.29

21121211

 

5557 6 9
”322$ 51..

22222212

 

 

026717.071.
81.088339
0 O. .0.
12211211
0 O 0. w
o o O
HumuuanVmu-L
2112ppzn
alhwnb.lhwao.l.1

Pods

 

6635 51..
11366 mu 8

O 0 O O O 0
314. 33332 3

 

8 17 1
mwfimufirm 0

3huu33332 3

 

3.17

99 h.
oranw e-le
.223, .31

3-35

2.41
3.18
3 36

 

71.1CUIJIJ
22607122

0 as o o o o 0
33333323

 

12 M.

4 1.11.
8 1.11.
12 N.

4 11.11.
8 9.11.
12 M.
Awerago

 

 

TABLE 7 Ayerage per cent constituents in bean leaves, stems
and pods from check and fertilized field plots
calculated on the dry weight basis.

_ Ayerage of
Mme Check 0-16-0 0 16~8 4-16-8 1mm
Dextrose
12 M. 2.12 2.28 1.82 1.78 2.00
4 1.11. 2.23 1.96 1.87 2.05 2.05
8 1.11. 1.54 1.87 1.4 1.88 1.68
12 n. 1. 7 0.95 1.2 1.80 1.32
4 PJI. 1.54 1.6 1.28 1.71 1.55
8 2.11. 1.37 1.53 1.22 1.34 1.37
12 u. 2.01 2.60 1.83 1.90 2.12
Average 1.73 1.84 1.53 1.78 1.72
Sucrose
12 u. 2.11, 3.14 2.74 2.36) 2.59
4 1.11. 2.36 2.71 2.74 2. 2.64
8 1.11. 2.39 2.25 2.52 2.24 2.44
12 N. 2.12 2.06 1.98 2.29 2.11
4 2.11. 2.21 2.35 1.55 1.30 1.94
8 P.1£. 2.08 1.61 2.51 2.61 2.21
12 u. 2.42 2.68 2.65 2.75 2.54
Ayerage 2.24 2.40 2.38 2.28 2.33
Dextrine
12 M. 0.58 0.55 0.54 0.55 0.56
4 1.1!. 1.33 1.05 1.08 0.80 1.06
8 1.1:. 0.73 0.96 0.90 0.89 0.87
12 11. 0,97 1.06 0.89 0.94 0.96
4 19.11. 1.02 0.96 0.88 0.99 0.96
8 1w. 1.09 1.02 0.95 1.01 1.02
12 M. 1.02 0.89 0.82 0.92 0.92
Average 0.96 0.93 0.87 0.87 0.91

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

CONTINUATION OF TABLE 7

 

Time

Check 0-16-0 0-16-8 u-16-8 Average of

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Treatments
Starch
12 u. 2.18 - 11.15 6.18 5.81 n.58
h 1.11. 6.01 6.70 5.52 6.25 6.13
8 1.11. .18 .13 . 5.28 .39
12 n. 3.30 3.110 3.30 3.111 .13
1+ 2.11. 14.95 11.09 3.82 .6n 11.38
8 2.11. 6.35 M12 6.39 8.146 6.33
12 m. n.82 11.55 8.26 9.38 6.75
Average 11.83 11.73 5.79 6.18 5.31
Nitrogen
12 u. 2.88 3.06 3.31 2.90 2.96
1+ LM. 3.27 3.23 3.31+ 3.12 3.21;
8 1.11. 3.15 3.17 3.05 3.27 3.08
‘12 N. 2.86 2.61+ 3.22 2.99 2.93
n 19.1.1. 2.96 2.87 3.05 2.61 2.87
8 2.14. 2.98 3.08 3.00 2.86 2.98
12 M. 2&3 2.119 2.38 2.65 2.51
Average 2.9 2.93 3.05 2.91 _2.96
Per cent dry matter on green weight

12 m. 22.71; 22.59 23.13 23.51 22.83
u 1.11. 20.66 22.31 21.78 22.30 21.81;
8 1.1!. 23.18 22.118 21.73 21.52 22.2

12 n. 23.38 2u.3l+ 2.129 23.76 23.9

n ma. 23.26 23.99 21+.31 23.72 23.82
8 2.15. 21.19 23.68 211.50 2 .85 23.88
12 a. 22.93 21.1w 23.87 23.72 22.98
Average 22.52 22.97 23.37 23.31:: 23.05

TABLE 8 Average per cent carbohydrates in bean plant parts
from check and fertilized field plots during a 2%-
hour sampling period calculated on the dry weight

 

 

basis .
Check 0-16-0 0-16-8 1146-8
Leaves 7.27 7.00 7.96 8.81
Stems 9.57 9.55 3.65 11.13
Pods 12.89 13.32 1 .11 13.05
Average 9.76 9.90 10.57 11.15

TABLE 9 Yield in bushels per acre of clean dry beans.

 

Check 0-16-0 0-16-8 h-16-8

 

Bushels per acrel 9.68 l 9.95 I 12.60 I 13.31

8.

90

10.

11.

LITERATURE CITED

Clements, H. P. Hourly variations in carbohydrate content
of leaves and petioles. Bot. Gas. 89: 2114-272, 1930.

Davis, I. L. Daish, A. J., and Sawyer, G. C. Studies of
the formation and translocation in plants. I. The
carbohydrates of the mangold leaf. Jour. Agr. Sci.
7: 255-286, 1916.

Davis, I. 1., and Sawyer, G. 0. Studies of the formation
and translocation er carbohydrates in plants. III.
The carbohydrates of the degradation of starch in
the leaf. Jour. Agr. Sci. 7: 352-38h, 1916.

Janssen, G., and Bartholomew, R. P. The translocation of
potassium in tomato plants and its relation to their
carbohydrate and nitrogen distribution. Jour. Agr.
Res. 38: uu7-u65, 1929.

Janssen, G., and Bartholomew, R. P. Influence of potassium
concentration in the culture medium on the production
of carbohydrates in plants. Jour. Agr. Res; 1#0:
2h3-261, 1930.

MacGillivary, J. B. Effect of phosphorus on the composition
of the tomato plant. Jour. Agr. Res. 314: 97-129, 1927.

Mason, T. G., and Haskell, E. J., Studies on the transport
of carbohydrates in the cotton plant. I. A study
of the diurnal variations in the carbohydrates of leaf
bark and wood and of the effect of ringing. Ann. of
Bot. he: 190-253.

Munson, L. S., and Walker, P. E. The unification of reducing
sugar methods. Jour. Amer. Chem. Soc. 28: 663-686,
190 .

Noble, Uber die organishe Leistung des Kalium in der Pflanse.
Landw. Versuchs Stat. 13: 321-h23

Quesumbing, F. 1., and Thomas. A. I. Conditions affecting
the quantitative determination of reducing sugars by
Fehling's solutions. Jour. Amer. Chem. Soc. 1&3:
1503- 1526. 1921.

Reed, H. S. The value of certain nutrient elements to the
plant cell. Ann. of Bot. 21: 501-518, 1907.

'

 

12.

13.

1h.

15.

16.

17.

18.

19.

Schuster, G. L. Potash in relation to quality of crop.
Jour. Amer. Soc. Agron. 19: 506-517, 1927.

Shaffer, E. An, and Hartman, A. F. The codometric
determination of copper and its uses in sugar analysis.
Jour. Biol. Chem. M5: 365-390, 1920.

Smith, T. 0., and Butler, 0. Relation of potassium in
plants. Ann. of Bot. 35: 189-227, 1921.

Stark, R. W. Environmental factors affecting the protein
and oil content of soybeans and the iodine number of
soyfiean oil. Jour. Amer. Soc. Agron. 16: 636-6H6,
192 .

Stocklasa, J., and Pitra, J. Uber die Einwirkung der
Kalisalze auf die Entwicklung der Gerste. Ziet. f.
Landw. Versuch.im Oeeterr. P. 567. *

Thomas, Walter. The colorimetric determination of carbo-
hydrates in plants by the picric acid reduction method.
II. The determination of starch and other "reserve"
polysaccharides. Jour. Amer. Chem. Soc. U6: Part 2,
1670, 192M.

Weiszmann, H. Uber den Einfluss des Kaliums auf die Entwicklung
der Pflanzen und ihren morphologischen'und anatomischen Ban,
bei besonderer Berucksichtigung der landwirtschaftlichen
Kulturpflanzen. Ziets. f. Pflanzendungung 2: _l—79, 1923.

Cole, Ralph C. The diurnal and seasonal changes in the sugar
content of the sap and tissue of potato plants as affected
by soil fertilization. 5011 Science 33: 3h7-362. 1932.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

    

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

    

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

   

 

 

 

 

 

MICHIGAN STATE COLLEGE

 

 

 

 

 

 

 

 

 

    

 

 

 

 

 

       

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

       

 

    

 

 

 

 

 

 

 

 

 

 

 

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

    

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

     

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

  

 

 

 

  

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

    

 

 

 

  

 

  

 

 

 

 

 

 

 

 

  

 

 

1 11 1 1 I <1 1 11 I 11 1 1 11417
HOU.P.79JIO.IWI. ‘IHIH an. H} OQIQOvoofi r‘ Ho..- 4 LY *fimwme WAd Hflw o It 4 a I.
o ..oo...__ w .. f» 11er tree: » +0.70 m HmHHHIA H 0.. T4L*AII.I v. I e ....e
‘Ilfieieoeeeau IOII ILVIQIOIIIOIQIIO 7.61751 1 1|4 9.5 l..r..I”. 91700III IIO
YVAvaveeoateoeeteoI ......v .enoeeee II Ieofi..— toHoo-I pl 0. r. _.u_.~s .o r
OILI.ITI:|+’._O>‘IOIO: ..I- s. Iose.<ooea A[.§¢ Ic..o_- .r.__-o. 0.. _ —. _. e n A. A... ..
VTAV eeoae1..._...... .I'IJIOA .OQIJ4IIII90I9 .» 074* u .4.&er ... ..A _o . aw . o . ... o o. ...e..II4I
YA oooecOJOe...I.l_F++II .0.... IsooeeooataaOo so on 1 I. ._«L F. + a. a. I.. e e m "aw ..4._IV~I-
IrAva 6690 .Q.. ILIOse vI-Oo .997‘19101]..7h+.1n» 4W. ...“IAI_.I1 F...P..H_.OI._. HA _ “an a ... _r 9.0 a a. . &+I. a IF.
IIWLYH.‘ 7010.. e O t. o . u o 4.4.3.064 IOIIYO ..‘IOIOJVIHI’IF 07w 0 (L I. WIWAYH e wF.‘Ia .. «In... 0.. 4L .. .I i... III. 7.. 1U. . “..I a. w w I
II» >11 _ ...IOIOqI. III I 1 A wiwr 111k Wen-Hi aIn std-J F%I44Lu+lis O [II II.._. +.~.lI.II.O..-I..
III H7itolboowen... vueeof seas-ew4ep «a o... 4'e*o a .mw..+v¢e.. a vecI<7I4e e u 14....I.
Ir AV .. .1 c . . .. ... 9 7717471119767 LtdoI .... e eateoiIfioHYsIeoeAL 09.7. I I? i e . I.‘ I I!.I>I .. . .
ITA Arvoeooow.v. .1. _. e. 710170507! fgoeweeo: .d‘ 4I_IA.._ ._04.u...l... I¢ u .eIIOJIIII‘IWII
11IIIIIHI7IIQIo0- J]... ....o o‘eo00c4 . oooeoua .. o . to ..e. I I‘d. _I;I. II. {oer-luinfl
LIIYOJIOIIIOLI’IIOIIIOIPIT... .- ... evanescea .thfvo-v . . ‘004a .a “F... 1 II I .1. 0+.PIIO I..I ..I
}1I IOIIIP: o u I... ‘w-I .. eeuocAt ._..oco. c o 9*04H ... . 4+~AU. A If. .IanIiIlg.7IIIOI.I7 FIT
II vetoes; er” . .o_ . 7‘-7.v...... 09.03.70." .+I+efebOI1I ..OL OIII +£..91.10A O7EI7L... .I-I_Irul II _
Ilfifi 09 e u 404 .. I . i... i 1.1.10.6-.0070 3-7 1 «I! 1 Pm. W 119+. r: 9:] 9 7a F.T II o .I.II.I.I.I .I
1119.... ._. . .... 1 s.IIIOOIIIIOJZvdIIeIQIToI..eI....I7leAIII AYH ItL TI .TOIfJHI‘VIO OOIoIIIQIeJ ..I
I q..LI—1.III _.I‘..nA 4.19 » .rII_ «0 II” I 11414qu41 I as.» ><.sllall 1.64] ..
H4 . a e . e - ... p _ I .c. I e o e .I o .4141 . re 9 .n. O .J _ ,, .e H .o H OAVIIJH... ...,AHflILHII 9011.7..1. c w_ ..
_..Io o¢porsau.. ...oo ...:oov e¢~eweao o!- IHAWO F34_Fo ._. e«~.+.e.-l_.n. .I .e a n_.a. ....
0nI§IuDJI7oIaIOJ1II b...0| aL vIstIJIe e: ....«o OO_ 0 .. ..%# .WLL. ohiaoue A... o o I 0 ~_*- ..
... 5... .... I . 3.1.2.3... _ .« ISL H134-.. 2-11-141... ._..-._.I. ..II ... IT. .
anLI+ J110.|01_I.J n»... .A +79...ne;o:ae +.e.~ ...«e . ataec. ....A .+ q n A JI: ...-ecvwwa“ -
>ofIa ._..... .a.e ......a..... a“... ...-e. :14.....e.e+++ AI .13 I..+..me-::ooI:II7.._. +e+t
I Iv Vt. mm.4_ e ..o_r tn... . e-.he<‘-.I60fi *OO‘UOOIWF .JJIFILAO {._.aOAVAIOWA. VF I”_ 4.0 IwOII IVJ4N_.+ .1
II vv7aI?tl.\Ol+.b.HIIFII+IT« .I.v§ L.1«4I9I...IbI._.Oe*eo__”figs..Fv. a ._ .... tn” . I oneflao t. o.II.I a.:4._ “-
ITAYV s;......:.. v.5e. ...+._.vv o.~o...:«.?e w..wo —I~W ....We_e..~.PeI$ ..._.+..40I:o1.no .wa
1111‘. Q : JitlI & . IF rel: 70.1.3671 II“: Fl ... ._. A ..wn. . II 0. I _I+ T+..I-I ..1I. IIIIIII E; III; WLIII JII eI
IIVAVi‘o no... ..I ..Iesfl _..r~. ..wwnovo..ee.u c O A; IIHIIHIJeIL FIFIIAI t.fi._.oI.#IIeI..IIII ..lverI‘I...40++ +.. T’90
01 Av...“ t» ... .e 14.: .u.ue~«mvi#oew..u. e. «. .I,I._IF+I.I.I.II ..90 .VIQIIIIhIo...b.4_v+I. I... I...
IvAYiOOIC to r e e .0_F __..w_Lvo ese+ I4! ILO+HO+ | O+IOI {I 0:, I .II .alfirs'vde.a’no
llleeooeH.o-A_Vevewo e .... tnefilfieoiv. 701T7. 1 A $04.41.wa feaw+I+II§Iv II .000. 1 H.._..m.IrOI ...II..Ie
Ditto. oI VOIIIIITI-I on... fuel 4.7w41 e 4_m e0 v07fl._:¢rcfi.fl+lveA ‘rholeet IeoebnosII ._I .70....aee
lIvOeopoe..fiI u .74.%I+A VI_.,7+III. wle..dI4_ H - -_ +O&.o.+firv I+_,IIF.I IfJbllIiOItL- IJOIQWC .....O.‘
LTAIIOJIOIIIOJ. ..o._II ... .- I.V4«_Ia_ ... I."_ .~ } a. 1.I7++.~_Ie ._[TrtbweleloenlIOol III.1.. I.... I
170.9,. 9 v o .. .- 0w 9 . s o . e e n r . _r a . e _ . “A 0 ..w 117.4 O_ Alb. wleJIIIA YLVIIMI e ..IOIfiIth- Ob ...IOIO. “I77. .+I.1I I 11 IO
UTA o I saw 1 . _ u . _.fl a..n__n .. w _ -* W Low.ea.+IeIeIrl..‘.A vnIHcII | a. a .I.. I4I . I fit. .. . , .
I ym1on r lltslll’l’s'JlfusAl4. . 1+I_—1 r 1.....4I- u 1I._III.. .
M .. e e ,_ .[v If. I III . ..I I III... “I!I7I;I1I.~I74I.I . IIIIA I. v 31“.... . e .. IiArfT 9),- .. ..I ....e e .
1H 1II.«I.0T¢FI v. 04-. 090 u....¢..‘«.. aAv ..F. o A -_.~q+- .AH.__ev _.«.n..
.9 10.75I9IPIc .IIIIII- IIIIP es aoI..|.aIIoa I r >III¢J . I _ a.m_a. ..av...evo he...
v.0 Are Av¢+_ .11» r“. 1&4 . I I7 I I I. no 0 eIo ..n ._. a +Iv L .1711 1 fo r .L? 1%.“ +._. + o I .+.,_.Iw .. . e v I I I.
VLAaATanI L74 v.1 Is- I IOIIJIQHII Vs. "IeIIlQnami .IOAT004 e. . .. . ‘.I_I.. ‘1..OY.0|..I-. IIOIII I .00..
T1 Io_t ” POI .8 o. w... e v ‘4 I o 0.. «04 F .l. .IQIII.I¢IHIHIJI'.I,IY are .._. c» —_. use.
..Aee,..In a. . v ...... .... a~_ouo.._. e.,+.4__eo- . . .o—.A ..
AVIIOOIIIFVI..IIOIIIOT.II.J so..bn .Iaee cog . +H¢ r._.«-ueo1._ To._...yfieo eat-I9 fi..o.>u
Assasee.edl.u.w. .. lauI I ._ . . ._.. a 4o.ro._ 4... F ... I+
b LbLIIIPI ._.. ....JI ..., II (In 4
YHQI¢£I I.’ olIa II III 1. I | .. n
1:0.0... I... .0. . ,.
uI.I.A..¢..I'II. II M I ...I o O T. I
.0: .0 II I ._I I. . v I .I_ IIIel Ivo-‘
.I I I . I ... ... .‘o
w I 0. I ..4 II . I l I I I, .0 IIIOIIII 1. A
II. ... I ._.. u . _. +_ .0 ....A
A _I. II I .. 14 JIOIa-ne A
. --.? .. . . . . a. grain.
.. I_._.I....L. ”I . . _ u . _ 1+ ...- ..14 1 1 . . . . ll. .114
I! ,. . I I I I: IIUOIOIIIIOIYI... ae+L¢HIr.I,II I _ JIfiI<DbAleA .IOOII.+. ....Ie_ ...vfie L
v ,. .r.II - _.~o.«4¢ ao.eoe.+Av9 0.0 “1%.,H. e co. .4«.&.I _I__n.A
. .e . I..IJ.. ..a «La... I'LLZVQ e ...+.. ... ee.o4.. ¢e ._.... A A
v “1.4“ ..Iena IIIQ..'I .AIIJQC I1...I&_ II». I .a. ._. ...IOL.. II. II.. .0 _¢ .1
ve_o ...... .. .... . ........o_.eae... .1. 6.1 e ”77.69. rvfece ..rc we Fri
r, n 9 I. t _. J .I I -I IFII I. 0.. VI.T4I1.7..IY.. .. .IO. aI _D.AIO_ .I. file 0 . al “:1 I 11,1va171
. 1... ...". _. I I III II en. . .0 s ._I .1 . .."T#I+.4.OI~0AYI 774 7a ‘ c. v. OISI'AI'A
VOITI. .I I., I s._ . 5 .4J .1. .1. .H I0 ..I’wifrn . iv . TLI.
r7"... . .I. s It . .. I I.» II .. 79 7,777. o .
III. III I ..I. . I I-- . AVA u. .71. ...?II ...o
'7‘. . . n _ I I I I 17.. IIA 7‘. e , .dlb II
C. I. I I I II 0 xii... IA vII 1 . Que.
_Iv.....I o I. tea]! I. 'JI. 6v .1
4. IOLI I .11 I I IItOrI .I . El .4 ”V 1
._ . I I In. I .1 III. v.1] _AI. 1.: l .0 +0. ..
._I .....I.I .II. o we.4o.+1 .I A v . i .5... .IV ......” to A L
7. 0., .I.. I. ..-. .... .I II n w +1 A F. . ... . Isle .I_A.I4e... t L
I: r. h . a. I a. .. I IIveIAIIQO AvrIA AYAw ... I {IV E. In. ,I..I.Y.VIIIIIID .. . .I... ..4. {HWHA Y L
HID-.....IJI a. ..w .I. .I I IIIO I I' v u- 1 )1 an 11 . .J .lll xiii] IA Ilrl PH
. ... .. ... . . .n . e d .....A e o A..Iv.I4IHI , .011
or... .... nu. _-V _. .I..o H... o _. ere. .w ...AVAAVAO
vw7I7¢I ... Ice-o . .1 . e .. I . l—e $.09 . are
Ow_.0. ... .I.I*I I u . . c . . eA II It .I I o I .o I. WWII eeA A AYATAV
.IIOIIOII.Oec| I6- Ififi o t wOs e .9 A V1171... AYOL "0 0.0.4 ., . aeA i 0.
vesa.o._a I .090... .0.4.I 97 1. A .. «LL .0 .IOA..I.7. .0 _rnoo alAAiLv
.IQIIIIIeIQIol ceIeIoI 1.... ._.oo._ crow- ._ I 1 “I ..m _..rou L Foe-L.+;._IYFII ~.
1160...... Q o4:..01).lu0091I+OJ o v t. .II ..A‘J+OO§7.;yI C‘IIA._1. We
weeeeec.. - _” .40., .I |,_..+Ie 9I_Ie...I|OI76IOIOe . ...._ V 17...“.74‘1 A“. .
I 4 III 70...... .0 II... c 71.71: 1+9 I YIWI 11.. « .rhIrIollbI _ #1
>1| o if so.“ I .I e.OIO.I.I¥f. 1 A o..- .ou.e._..IIJ 77: AVAIAA
vwvoeeonIIoI. o... .saoaeo. I. ..a.-.. +.o_. IIIrAYMlI
«.41 o: ...... ooIr. 10+..e$a+o.veev.... r. v I..T:F..o.§§.o
n. 070.. ”a ..e. ,.I...e.7eseAr+§LroooIe Ir 7+.w.b.7s._oee. IocI I.A
.... ... .-....q.... ...w .0 T+II U . I»...8e.o..+...«:..e ._I‘I .0
YA.AbIea.I I .. ._., 17.703111 r +eeo . ... eeeoitIoooL I. o.
o_ .vexJ. «IJar woeoaeiIoAYYeveleI. . Au .9. .a44... _.. fur I»!
LIIIW+65297I ole oo+_.7% IteIIo4+AYoOATI9Iu9+OLL IA Awete. are.» U ._- I
- I... 11...... ._-I... 29.2.: ....11. L ..-n .. a.
-e-»1._.r-e.i I .. I . . I a I- .3 .- .
O.IIAYI,\IcIo.7.. . ..._ .... vysiellIfiHMIfizer. new-e: . ..H ..a. $7ufi7 w H W
tAv..¢.¢_ .,._o..7o£daocve... M..afi ... r IIITOIQ+AF7uoo 0 L1
.1 Acevooea+o a. .. o I. 70.31.? 9. o. .v .ceo Io... .....v .I ...I
1 Olbtlv +47ILITQ'IIO7A cocoa... ..e ..o 1L...IATOI.IO .51 e 1 17A;
Y. L VHAwplIoe 4.... 47.07.491.771 «49.! ”a” ... oeeJaoeeU 9 70.91171 9 .A
I Irv Areeonr .. I4IDIOOI4T.YI.7_I.QI.4A.O Ita1oeev .“ 0 A05- IIblgoae'h e I... 100111. ArA
TA I F 0 ea. eeIIeIoIOtIIIOOLqfleio— .. ..u. I 1 DIIQIOAIOIOIOIVI+II+OI I e v..004 o¢ A
I T +Avfleeoa OIOOIO.O7I oft. Oefih 0*. .0. 7.0“ QOIO II0644I+VI . I _ . . coon .. 4w AIL
1 TA .9:..... I oIsIlIII. CIT—l eeeehmee .. . ._..» ... ATOIOCA IIII ed? a... .0 . ...rIvIa 0....v v I I. . a. o 0.“.. ._aAFLAvIIIAI
. w 4? W9 l'uOlQO-i I1F4f+1~ _»|I1.1 MN} 1 ta 4? _ EA IOIA Joli-O‘CIOAWHI L I'VF'IO IT4O¥1.-Afil§t.eln In. 1 “10 1:
Iv a o eI.V.? .. n. .. ... .o-._. ._..awae‘lrlfi. OHFJ41 I #79 14-. ... .« ......aTI ,II.:. 0.0 I .III .1...m n... .... a on .. awefi «I0 I
1Y6¢T>¢I+I6Ioel c577 .. . . e no .I._.+ .9110 for...» o L newee.ouowu_. . ....I 1I¢.0I__.100s4|.:: a e. «a. .F.. ......o e .F .AIeIe .4 WTl$vaeo AAAI
TAvF*.O[AY+II.eI ... . . no . e. be.§c.1e.oHe..v a my. to no.0!lIA II_.e7I%Iflee vcdu. . o. g. .. .- .... Io .IOIn - .A o... .....A AA
Irbrfitt.7 to I as .. 1.5.80.0;IXII I :.«I e .. .oeooooLcI III...vono.¢eo..oo..IH e... . .. . , oe ... ATA
IthArdA .7. ..eoueaeo .L II Iso‘JlJeIeg .4eeoua o 8. F77 .ueI_so.cI.o. A76»...~.oeo 40a.f.u. ..a._ .ore— e .IIFo. Is, . H mo ".0. 11m
ve- o a... . .. ... v ”I. .v>.4 ... e coo». +.0 so. are” ....e 0... use... a... o aIequIuIr I... 4.... e.~ v.9..AoIII
FOIAO 1F. . a . 1 :JA . eem....¢. :_-.e IA?! 1e00'1n+ps eo..I..oo_e so. ... If; 7. 0 I
m .9. Its. .=:I— . I». bur+.ILJo..:meo. 4v.oe:ee;..._ _. 1‘... m. +. 9.0.9071: .
r .7...I . _. .I-.- 37... ...;d. .L..... ._ .. .._L._Ir .23.: a a. .1... t
4 FImw H1 ~_ 49 .v+970>wh1 $5.41.? .«+.|.u 1 A. .. d F1 ... .QAAA III
vTAvI rows. 90 IoA4TAc.«.o eAHHa . . _ e.t.e . 1 so» to
0 __. v. .70 fifIOWIoA 0.. Quote ....a-vewva .. oofi zrvfiw
> . A70. .3 e o 0* e v.1..4le uanbIncIeIwaO . .. o I}
«‘4 .e .“ coowIm eonIIIOeI..IOei.TI~_.e-.040r Is. 11H
0 e. v. 4. eaeeoo. Trifve...’ fist-..-,a ... 9.901.1w1
AIYre++ .9 be I..+.I.O.o.f-+o.1ooe+:4 H..«..._. .1. .11 41H
...IwIOIAY 9 “H09. Anowoewo. JYIIefiwé es «1 o‘ AATAYA
£1.11. 6 ... .L. .0990... stole . e.. '1 $# I
“111 J¢li9$fll0+ 1|!- IJ. .oIJP
« .I .4009 0....cveo. ....e... ... HIOLITOo
¢ aAvAv’I. .t' 9.0:. ea.»ooeou oaaATooa ua $IATAVI v.r
'IOIQIHIOQI4III‘A.¢¢¢OOOIN 9.40.“. '0 MM oflwfi
1AA sooeeA III.IOII4III. lirYAIO. . fie. .O o g»... M IMIA
AYAII . a u o . - +- a .. o ’0. veaekfln I7:I. III...OI. .«c ... AA
we . e: to ...—A... .O‘IeloIWI H... e. ”so
~.. .. ..os...o.. .n EIIrtIYo .n 04F e>
. Iiwenu .. .~__.. ovo...o.«< It, . «mFQ v
z r iii. Lb..." ..I.- a. a

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

 

 

Dt.o’ARTM ENT OF MATH EMATIC‘

 

 

IIIII

I
I

 

I

 

I

 

 

 

 

II III

lull—I

 

E

 

 

 

1
,l

{I..
VIII

IIIfI
III

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

MICHIGAN STATE COLLEGE

 

  

        

 

 

 

 

 

 

 

 

 

 

 

 

 

. I. .
In_ olri

 

 

 

 

 

 

       

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I...

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

    

 

 

 

 

 

 

 

 

 

 

 

u. M
4
vl‘. A“
5.411 .0
G 0
0L o
”.1 .
w” _
. , .u.
v o
I“ ... 0%.
M ...
n...
. IOOIUI 1.“ .
o *Iv+ o _
.6 ..QII..7.
._ FL.
._ n
O :o
J...
o

.,
...I ..f ..
I.“- I 1v-

.I. .04

  

.99...
It“-
I."

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

.r
p

..
..fi

         

o

.—-._'a-

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

   

 

 

 

 

 

   

 

 

 

 

 

 

 

 

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

7'

1.‘

_.
4L .
r
t

. t

.
0
A

.I.

 

 

 

 

 

 

 

 

 

 

 

 

a . 2327.....343.
_ _ T... . O . t b.
- a on g a 70.04.
... O 9 _. 1 .. .4 4‘ I.I.

. '7. OOOOI‘. OO..+IAV4'I.
00...] OOICI» OOHIC 00L.
bI'I'IOI’ I “:41. 14 9 OOPI’J‘I

[7061 o v I;

If. ...I. 7.qu
J I.» II; ISL
. I ..$c71|7'§0.

A 7H .. .7...
IOQA 01-1 v I..

I

 

 

 
     
     

A...

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

.. I. A. _
.0 . . H.7‘l+ ' P
«._.. II @II . u . m a. . <._
Y. A. _. . :1. .I. T [I.I-7 I.I‘ II —. O . 75 7 _. o - ... .I. I I. A .
VAC—.OIIA: . . I-OITI4I III-'97.... 4 I 4]le III 0.9. I I filo}... I'D...LIO ,v
...,P+IVII..H . v. ..‘IIIIIIIIJIQOIIII700 . .IIATA - . . a. .o .. I .q ”III? . .I c
+Iv>0an...... o .0 noto>oc 4 ’ .». 4IIIIOIIOOIO..oI . ......01 no... u . .
bIA'roo->o;,..+ to I Ilili. .. .. I one» . I c n I 1‘ .. r ... #
‘Awi I60..7J...o¢l.l .I.. ......‘IIIIGIOA ITO” I 1I'I7 o+oat 0| 1' . .p “tr I. .fl . v A
IVQO-uu‘;o_ _ o. .. .IOI7OOIO .o , (I. >II‘IAIflI'IFI _ p F. I _ 7 o 0.. n u A
I 1I|I4I0 I..... I..:vliIOIIIOnhn _ JATAI v. ... O_ . 4 A .I _. I....iI. .._I.I.. 1 "L
IF; 7. ~.r....v..».._ . ¢o_.n.c..oo.. A A. QA_IIOI IOJJIII . I 9..- . .-.-... F.o o
A D DI_FV¢IDI_ '5'. I! 0| 7:. I ..;.. ’FINA . I.. . .I... ._.IIOA atw v “.... *A
.I.... ALI 1.1.1.4.. .. . .. -.--.-I - .... .1... .I . A , I.. . . .I I
T I.I I 4 1.4. 4+.“ 4 .0. Q.nl.-l'. LINN-..llihlil
+I. . . .Ldrlwi. 7 w II Q u. ._.”.IIF . .I_ .. : , .I. I. .I I.I! .LILI J. | I A_.‘DI:1.I.IO.IA
,wo.7&-. VITA 'II?. .M—|._.I I. IJ {.IIQ . 'W‘I ALI... ,..,.O,IaOsIOJI
00o§o F+F . I. II .w. Ilon.I¢I.L QIIOIIIdlIfII..7 but u .0....-
H....66. .01... '00 .utffio A .50! 0 7o 0 . .. .‘c.
o .0.60. .....4 I Onllw 4L. . ..u.. .r.. ~Ir¢1I¢1I .I 1.§fl+.u+..IHOo,0F otoAL
.94 99.6 L V}.... O 064. W1 I.I 1< .I,.I0...+.,LI... ODIiIIAI
0A 4.. 4 .. .o 7. '0. .OIWArIIIt 7.. ......anIQiI+ 1.1.11
9 VOOOOA—: HA 0. ..fl IFIOOITVIOIOoI Q: IIIIAIo 7.... 44.5. I
Q. ..Ifi . ,. . A4 . o g. . . .. .II ATVéITI Int IwLT .HI . I
. .I 1 .. W J 4 lo .106}!pr .I.? . I.I .II
vA 0... 0 II MIDI 9 . .....Asooe 0*. .¢O¢I¢¢760700¢A
.tA uFIO. VIA I I1 , ....n,o..0-oo7t IO OJ 4
...Av. ..Iofio 4 a m 1?... v OFrOBA u v o . . ¢
I I I .4 ..+ .0701 In 70‘0;
I A.» H 0 W4 A. 794. ... y.o.o ..HIIJLIT
I P 4+ u? « GOAL .«I. I IIIIJ — I A
I 1 .7 us ooofiIJWI. 7171;“..on . o H*. AA I
I I I .7 I LI .... _ . . U
I 0 4 II .4 v.
TAAHWOF ._...fl. 4 o. « ... .. “I“.TIHIIOLI!
Th 71 0F... F . (CIVIT IIQIA.WA 4* ._ I
IV 0H4 0F... tut. X .990 OI 4 I. .1 vooo‘filo o. uFVIOnIc I
v d t. O «I. vl IIAI. 0 00 o I 4 I 7717*A . .w v
.Y o; 'atvo 5?... YOI? I._. Ildfii40 III 4.0 . .. on.
Irv“... . ... . I¢n70voH0a o . _ IF up.
IVA.._¢4 _ {V . +. I for. 7706‘ t I
II“..- QII H 1.?on 49!. 0 I64 0 ._.-
YA LHJ .oo....l.... .0.. ._.-.4" .._..
v I .II’O OnlI 9L .. 4 VIII. I. III-l I ._.
0. ._.i ..Iwa TFQII ..I... .H I.. _ .
{4 OIOOOOII - I.I.OOIOVIYQ OIOIIfleI . ,. , ._YIA
U. ’Ao F.7cca0.‘ act... . o O . .p 1.0.9 .u I. .roIOvIA. ..Av.1¢
Ava. 4". 04.... ouf‘oo o .o t. v u .v. ‘V. 1 cII 7d.% 4 w
ITAIYAVAY ;oo..v-. ” +P+oio 0% O n a o... o ... . .d cri
O‘ctaoo 70900. 6 0+.u-Q4 F . __ no; r L .. cFQ
o _ o g . o I 4 0.40 70 4‘“I# 907 . .- .fl. I1 Tull.“ If. Fli. LI... Hlfnf.
Ia+. .9 .IF .. I ITPIOOIOO rffblnerQIII . H .00 0.1...O06I ’F .v 0' voooA
'o V . .. . O o o c .I.. . . 0.. .
I. I...t.x.,t N ._. N ._ . . 3:: n3. ... ...... .
HI 4., .E... . a”... «A .H1 I J I 414 I
14- .fo..«0Ia,YO. I 90. 00.0 :A . m. ¢¢7A IQOIOQIuIAwIIQIpovdr .1
con... oc.u0¢..oo. 69 k .0.. o... +( o. AWIJ Db:- IoIQIA L l YA
WI«94..?.ro.zo¢* . coo-.I. H.... . 0* .4 a, .....I7Iv .AfiLQI VfLI
..«IIIA10L9..79I_.9 . v ......A .... . .. . ... oo¢¢ A +« .c A
H0§04ouaau7HoLT ova... .m. a . o. ...u ...a .. .. - vat:
o.¢v.4..o.... o.__— A- “"00." .... o9 ... .1. «new um. cc. A
II. Ayo. ...... 0.... Q. .L o . Q... .I. .I.. IoIvIv t...4wo.FoIvA .. LooIAA
ILTAT A o..ru.c..- ._L A. OOTOO .....I- 7}; I. 0.1- so a . .oIVJr. un..IOA. Lvowfio
II T “engnmoa..o -” VIAVIA OHIOIoa . 97«a.. “V Imlf II7OII. .If...Oll ILLL . I0I +._..L
IJ‘IO.IOIOO' .0.... IIFOA I LL74! .I.. L 19* u'pIOIO '0- I III I III 0
IVOA 7?? u. o... I. I4 +£4-00; ”1». H . ..o I 5‘3 .IJJIOIHI ,H I
”IfLAJ.._...A.OII+dI I.I.Itruio v 7.vé.. IoIv+QIIIfi >c. _ ... .. oyo .. Foo ._afiow
IIvJAooIOOoO.7¢OQ.Oo ¢’4 A4... ....“ .A_. . on. .9 O... . Ir. ‘0 o...
ITAvOLVOFI.I.OuOIO¢OI¢OIO.0.A..*I I! 0A.: A I u... ...Fp L . .0 .00 0 F0 — .OOTO
LT. .40....” c+u-+. upoo 000- , _r '0 .lffi I90 ...—.6 I I
IIIA§¢OOUOO+IOObcIJI 709.? 6A .cht o I.. fl... IF. I4". (I OAT.
.I....OO..,40..0.I 0 ‘71 . .Lr - *0. . .t a .v# IOOIAFA
¢o .o._,,.¢... A... . . . ... ._H . . .. c A
.Lt..1.l..?.. ...._ u . T: . w I.
_o.I.._tIO_o - I .I. I v.0! Ifiur I00 I. 4
oo..¢op..9.v-Oo ... ..oIv..V.OI h...* 4
Ituitloo 17-- . ...... O4. .
o,...00.>.000. Io... 6.77770 t +. AF“ A
fiTw +‘J‘OI. .IVQI I.I‘i A III.._”IC ... 05-...
WH_rIH+.O+R.IOLI_f.OO¢IO ..11.0 '0: I.I-OOIb_.~
. .IOIIQOO...vI . JFIVFFIYA I 00.7! #1. Y . T00. Ii+ I
.1». I..... ...: . .7 . .. ...
._AI.~ I... IJ. v17 . 1 9. I. T- I..-1 .
n. .H. . .I.? .... I017“?! .OT. QIIIOIc I751. o ... 4
h. OII'. .ofllQul “to.“ 1 I 7!- I.. ,
AI d-vaow ... «Todm I.. . 0.. 6 .
II. It? 0 . ._th ... fl
AH. 4 .c . co. 3..
A I§ 4 .I+o.4 ILIIVI HIV-.OIQIKLQ.
AYHO 4 ...... ao—w- _ I79.
:0 . .TO VOA I..... «I7.
: .: :: I.I ...IIII

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

DCPARTMLNT OF MATHEMATIC.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

      

 

 

 

 

 

 

 

 

 

 

 

 

 

 

      

 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

      

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

MICHIGAN STATE COLLEGE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

       

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

  

 

 

 

 

4 . . . ._ __ H ..1. “1.”..-
. . . .. 0.0.9.. ».,.>. u. ......lfi 0
a . 1......- II
1. _ o... r. 44' I..
.o _. . O .
h w... ._. 1......“ _ WM w o
I | _ _ ._.... «out
1. 0. +8. .. f1. -
v ...-.0 .a . co
.fi 110 u. -+I_FI.. Y.” ..IMI. -f-O II .1 .
I a . .. ..n” 4. ..r _ u
. ._-- ._ ...... I ... .
I. II ... ... .‘9.. h
I .— - +0 1. ..100 _.o . L
r. . n ..w_ :4»... .... _
_ . _ . ._.... . . .m
H, II: I ‘4. f .u . . . .
c n a o r.. a ' .
1 . .I. -.-I. L 1...; , +
..I EL I|ILI..-.¢L . ._ O
1 . (Z...- :» ... a ...-«He H...
'0 0.0 “.l ... H . n . C v 9 .— 0 c w I I . . I.I: ._ ...J .
9, .010 W a“ o *.H“o“o ' w .w
0.... AL..._W. v........ a. ,.. r
...-.00 .= o .u. Toffi... I-Il¢11.a4.. .-
Y. .... . 3.49 .17... u.
u o. ...... fi .0 __ ‘\0v 9.0.0..-. pf. ..
... .. o0. .. ... d..+.~10 1 .._..A . . . ...__ . .
.1 x H 1. ,WU. #.I.\..I.T.ol.l
Y .._ Yo. * ... ......“ .v +_ fl. 060.91.956TFL.0...11.
11 ulml .... ‘ H¢H>.Ww'04 4v .4 .01.? ....ll. 0.0+“..v II.AI'.r.O.
.1 4 1 , u 1 .. .1 : l 0“... .06.!9...
I. Y..I4I *ovH- W*.v. 1. .4 h! l 1% .a-._. ..-. ......IY.Q.§.I.Adu
v k“ .Ifiub‘l H11. 4 7." W 4‘ 11 Hlol 1 I Illnlldl, if... .
. O 0 , ... . .1 1 I ‘ I'll In I. . .
M O ”I 1L7“; 0 .4 fill} . F”!.I.4§r+ . ..L. .-.-cu
m. t” m... :1..- E IRE” _ ..
W ’4 a... 1 1 H H ’vpfro bl.
.fi: H0 1 4.“. w ...!IUO. . ...
.... .. .MYOJI W. .J. 1 . m T. o. .IH .I.
. ,1. ...-0.. 1%rhw. v.v.$. 0 co... . In.
L rlr- vtt. v.0 ...r0 0 01..» I..
1 n._ ,._— vs .H..oo. ... a.~ .1
L 1am ..L._VI~I¥IO.VI.vIc 090.40. .5 o oo_ I.
v .... . 1v ... ......I u........
OI It ..WVJFI’.I . LIOI. I . u... A ... I IO|+LIMIF . . _
I. C... .Qad Q In..4 an... _*_ ,
I .I.“..O. ._->_._u.o.eo 9...."weeau... PI.
I .010... .%~..H. u...f¢.¢.0.ww9.0vw.n..j_m. ..
.4. r I'I.I.I r . Ill .F; .
V. chi 4*.ho'+’.b[.}¢ 9.. .I_~.QI___HIH .#
I vm... 1 00......00 a a cat . .J .
.I....“ L v III... 1.01 l.4vl1o .lxr I I9 I-I .“I. .. $.1le
I .1. l..v If? OI. oiil...l I.. T... ... .l
I 0., fi.ooo co...fi. b. u...“ 417.611
I .oQg ...th‘v ......l. ._. 0.7. .H
J ..wm. ._ o...vovo~+«..‘ o. .AHI
I WI.-¢ l . A._V.IO-6.— .I.-.. 0.... .I. .f..'
7. ..9. ..LI 0 I. .VITn..r+I.G. .H (I1 I l I.. . . -. .. x— . w 4.» WI
i . L b. .
.t. J. . - - . I. E1 - --.. .LMrhw. « 1
H H .....w . .w.) o o o..< .u n .n... 007004407 A..vrw . I.ILI
.. a «0‘. o- o o... o uo.s¢oovfio‘co.vv.v ww. .0.... n
.... . ..1- 7.7.... - .-. f. - -- -... TI...
.._.+. I... ......... ......oaco:c+oo.1417 f7.v~vo. rolvfi.
HI.“ .0 . . 1 OI..I9.DI.H.O (l! T... .. 476.10.411IH 4 5 9... fl «J p O.Vt
m. ...L- rOOv'ov‘o 0.00 >4 H 04"... ++fi1 ”00.10..
..[..o.l_-¢ 7...... iv... 0 {c.9100 4 w ..r1 4? ...: «J. 4 O. .5.
I .11...-04. I¢l I. ..l Y: 01.1 1w1114 HI+ v.7“..H 0H1. 00+H «o. o
,t.._f_0.o.. +. w “_+ -F > F I r» > t _ -w .r y
.....r.. u. H. . o LIAM. 1m 1 1404 ...A . .w... .
._ o..n..~’-.Im +9 “t a IYW.,“I+.161ro1.6o w o , 1 1.10.1
.P 1 + H1 Y. .. ... .0. 0.. E: W.1.4L +7-:¢0 . .1. . . . .TO.
0 h. o n... ._. H4 1 19.-..Ib0n .wfiv..¢ It” [.I.]
T... ...-9 . ..t d 4 41 114‘ . cV I‘D: ..
._ o _ . ._. .___- .0 HF. .v 00 ’»O. TLI Ivfiotu
.. 4.-.... 1 .— .61 .H7 -oo.+...v ‘7“ .1 I 1
+6.4-¢HH..W: . Tfil.w_. ... o. .oceo... 9... _. [1 .
h1..., wL. .. y. .w TNivWO. ...row. 9.... ..
4 4 . 1Q 1“ .I.
1 VIM'fiIY» v... . 4‘...“ M 93.9.1051} 1 +I<d4o$ .
u e _
9f.v.9 6 CI! . o ,. . _ .- 9 o a v. 9 9.00.0 4 9.1L. ,. .T- n a o. . r
.r 0.. .e.. wuw ....v. . 4 1 . v. mwov‘H.11le. 4+. 10. .»
w, ......o. .9 v .p . .+_¢ 9...... _Ho .....a .. ...]... -0 9T
9“ co... ... .- .. SW 4.. v0-..00IoIoQI +11%.¢_. o-“oooom.> - . .9...“I.-v| ..
o 10.noo [.1 n I'LOuf +Oo¢won¢uuow5o._+‘ h» 11 4O.OQ¢._ ...p.
o. cod“ -. 0.. TLIT.§_.O..OL...+O.vv..oJo chafi-wwqwoo +4..OL.LV...T.?OP}... o...
f04r+tau60404r 1 W AHOH'Ii‘VWC.FoU>.§b.tb +Y+v-vll. .IvO.1IAITAT 1'4L .. b. u
t! voI.t ItoiII- 1 4 t.$_.¢vo“ o t 1. ‘1‘. _ Io l‘iaIlH+LT .l-vvw..o H .4 ...a a
.. _» . > .h .. _ [IL .tbol L. _ r
.. 1.1:“. .I.- Q I ...I J - . ALF . o
u. .- ._sz .I.Q*4 .9H" oooct .06 no v09.0.4*. 1. 0|. . ......I 1
0.o._e4- .44._r.oll..0104.1..00 +.o.+o_ . ug+fim 0...... 0.. v. ... ... ..HQAUY.
I .0 n A .+ a. ., .0 v. 0 a o w u ‘ a H. o _r F F O.A.. . P'I ..I I.. +9.! _.I. HIO.~1 Li _ t 0 “4. I O .YLY.
I 0.0d.4..o u «a r004§no v w.»0.&+ .0 bLfi- whoo§ioud “ 00“ on .W.Y.
I V ...O\. .I l .0 00 O 10 4 V. uf~.k.¢lt.n..¢.b T. .1. il.. 1.0 ..1 + . w . .— o o h «a OI II
J ...-....v rocoovoo ++._.¢..o. . ob .MIflcé 10.04.... 1 ..r- o+.4 .v0. 4.” I
.LIOIQI.‘IY, 410-1.0Ia Hflt «a... ._—§. ... ...... to . h .0 .4... ..6 v. r Adv.
I o-wtoo-k 00.0l IOOOr IMWI.*I.I%.IIO.OCO __._V;mt¢4.~ .o .H a.; .. r01... I.. F1
_. .. .. , . I. » _
I .33: W. .....3. v}; .43... .........-.L.... .... ”.... ._.. .... r“...
I. “-... . _ . . ...- u _ ..r.. .. -
I J.M..+~u“4 ”a ..Ifi cormm ._ .N.vo *NAIII
I 11., art-1 wavliyrno.OI...r.L[ ...... II
Y. co...rc-_4 4.0..r..4.¢l..4_o9a HIVoan
Ir 0. ”c . HIYVI...IO+OI0.. 7*- . _
I 0.4I. .9 F a ..0 hr- .... ulo.¢.¢.o W . i
1‘ 0.0.01.0..- .00. ......+. v.. ... .
I «Ii- 0 . W43. .0 . n w A. c s o . 0 _. o c .. ,I
If Y-..|.u..sul_,- . ..OOI .c. v...¢sp.+ , ._ _ _ . .. .
1.. . . I .0 y > _._ .. W 00 .fl _. .1
__ a - . u n .111 +1.1... .79.... .I.-..- .0....
v. a. _ .. .wfi ..1._..o¢.0|0 «« o. .0.. ca... v.0 “ at» ..fiQLv .v..u.
Yb 0.: 1 I12“... ..-.w14 I.. 1'. ...m.- .000 to. .....- .....r. .0. o. .vc0o0.¢..
YO v.1.un “l.‘l.1|.¢¢.c00o .... rr _..¢ ..VCI OOIOIJ. -v.o_... .V‘H..._o-.w..l .... _, o. _ o. u.
v. oLLc-on IlOIF.¢¢¢ec-.o¢o.¢ 0.0 ”a. ‘00-... VII“. 04‘.- r % ... +9.._-
#0.. . to wet-...... ._. .0 otfir T..vI..OOlI.IIi.HI.. 0¢.r.v- u ._.“;
OHiIQoJfi, ,.v4b+0Hw $9.... _ 04-... .,c . 1.. .o 0.... 0.0....
o_- .. 0; 1,100.0 OI.I‘II%III\ ... to... t; . .Ohtfivo .000. _I
u. -.s .v «674* 1.1. a... awou I. a-’.|05.4 «‘10,. o . n.
1 . .. .1”. ... _I :u. ..E I... .._,_
. L . w I r
... ,7... I.. 31: 111.. I.I.Tt .. ..
.T‘ 0‘80. ._‘I-.. o..ootut .10. .. o ... ... , .— .w...’ HAIILVILL
.uw..-1 '61-... rH .e.....400.¢aw+vocorro o-aoc_v.. .- ... v o t ow c .H..]].H
* v..... o .0co . Q 0.. .n 00. ..l... 0.00:0..'..'. " COO . O a .. .. W l. . > ..
'le... c. ¢¢..P ”I” Hrlmlhuk‘w 4<00.Q0.._. 1W...;tlrllt.d|..fil...0.;. .H tr? _0 u”. n' a“ 00 c. 90.. .I. ._ ... I . PH.III
.0.... .F _ re 1. _ .. +._ ._ I? t. .... ...-Y.~o.95914
I _. o u .0..- ..9o ..14 .. o I... .000. 9, . 1. I v I 9 .
H7 6%1‘.>9r f. HIIW.#Q& *0. VV0‘+QL. o... . .__~. ..AHfo -. u..-r+MOOFU¢_+u . I 1”.” r.- ”“I. H .H” HIHIHHOHLflnOIHHHiv-LTI
..Vof ‘0 kn». ot.~.* .H ......o. ._...n59 »_.*? TOOHootw.L&AHI+.-..-1.1+». .0 I... n. . .. ..w ,w . .11 ... .
VII.» .I.. 1. ..—4o. 0 _.uo The: IH‘r...a_oIt.O OMIJ.I ..Y.Y...Lc . . , _. .aoo .... cn+ .... ...moco. ...I
u 1115111. -4 n ._.. 1....J»1 «11].. TT‘JL o .119 WOLwi‘l'f‘ ...-$1017-17 5.9+.vwncflll LLL IQfQ.¢.r-4.++< 1.11.1 4
_ . ... «.3 114...... q... . .. ..., . ... f 15.....- 1.--: ._ .... I.........L...irx
0L. .ur....r +¢LI.4Q. fi0t10o¢n ..;+f+_.ch o _fi¢..>”:-Q.1.O.o&_.~.-r v0 -‘O.......f_—.U ... raw. c . ....vco 00... erg? q _ o. u“ I
r. -....D..r.-ll.> olitluse oifioocc..... ....oo. o...m......-o. . . ..- . ..o ..._ #.fiv.. .11I... .0.. . ¢.u.fi-H~rH10?.v.vv.I
r M .t—O no+.vo“.rf000il"!....+r -w.....9 ‘..H,_H.,¢. 91“... OmarIwWHa . ._A ......1 v:o.‘n_.¢1a‘_ _.a_w¢ ¢ 0... 96.1.11!
Yfpicbltrd o. t .ooH Oro.o%?ou.rio- .. 0-...I.1/L..n< -\._.._, ._._. 1 .u—«._.. a.._......__.w ... . .. .H o(c He -* v1
. + . L. w. v.4 t..+?.oo $0.02..-. {audacw r-ufi. It r... 4IT100< .I...- ...“4...-R. a- .50 ..1aTI . CooLIQ. -.e. _<_ .1
F... 0. .oVIQIO.O:lor.b....I...I..,..r.‘ 0000 o ..u... o ..“w. 0.. -.g- ... .ofi...0l_'. -__ . n , w...... o ..c o. A
1*. OHW Tb o*‘.o ...rIHo-Ib .. I I VF” ... __...: uI__I..|+rfl I ._n 4-. [4| ha. .,*_ ._m__fl.. ...II’, .... u ..v. vv; . on..1uoo cc L A flv
”a...“ v o O - .o co... .0.-... a. ..c 0.. .... .. .H ._ . ..fI oLr- . FLA. ... ... : .. . _ , a
F H D »»»> a up a). h. % . P.Lrh I .I.... _ b ¥._r __ .r It .- __ ’01... .- . H .F.P.n ......oHHoOww oHH H

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

DEPARTMENT OF MATHEMATICS

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

    

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

    

 

 

 

 

 

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

MICHIGAN STATE COLLEGE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

    

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4J1 4 4 I 4 1 <1 4 {III ad 4 III.
4 .— 4 i 1‘ _ L 4 [old/II. .1 H H M -HLA < _v « .... .M; I.I-IFIWOI .L.
IYOOQHvo H.«-. _ INJHo. ._.. o 1.0. ..Lv ~ 0,141 ._ . _106 A HM H I.I vTITHILLIwIHIHWI:H+o .... «A. ._I-fi.r he . wt?»...«_.
“Io.o...”_..4I0qLIo n...T¢qII ... . Er. ,I....._...HH VI. :41 .r, A H o .LYfLI+.II.Io.JI.I»I-. 4.. c _._.H.. 1 uJ_-_v_o..
Ilv‘obvoo~ca.IJYol ... c. I-I. co ._.vtjo o 6 vIHIiIuliquvopnIurlvllv 4 <mq... r 4chtl..Hw LOO... ....H...V+.Ier.. Axum-.41“. gL .‘_oa_.
11.2.311331 -I.! I .. .. .-.II ..I; I . T-IIIII I - _ TI....LIIL.-..I ..__.....-.-.+ tr..- I-..P......-.;A 3.1+.
IIYCIQIn O O 0 _. I - a c a. ..II, I. u I o . 1 I! glolo .I-.” oLVHH oleolI. «I! I.“ ,OOIOIOIO b » o M liHIv 61.7.. VT p..9|+ #.r r a v 0 q . u n o O MHIwrfl LL-I. filml’o O+.... I A. A _. ._ o . r O
11....vfi+fn IkchA‘ to... WIII och . Ii A _ ___ h? A. > .0. v .ao-oQOIdfiI .1.,._.I. +IvLIIonLIo-9Ihooo. n _r.9¢.II I.
r._~ -. .I....v, . ...... .06.....150Ioto.¢. v. 00.1.0.1 I I t1 .~ +_ v.*I II 1+. _ _ __ _. -
11 aIvIQ‘IOIIoYT 9§ . a _ v o. w. . . ., Illi‘b... a n :A A Lvo
DID... 7 _Oc .lI. . .. 0.. . ‘.'I IAT'hleOlrnJIn OI+ $5.... ‘0; "a 91. #r \nHv IH q -_ , ._..voowv‘fi.v.I+Ih villi
I ......oil. ITO-o). .. .,.!.A.laoa... .vonm...F-H1HH H I.I I§_L.o ..F I , .r . “Hr L P ..I L .
3,0,-rirwll‘l,l I ILIAI'IJ'. I.I'IIO I-‘MIIOLVI.‘$|0<_ H 4 _ 1 .1 . HHIO o, ofilfi r . O'Olal .9
.I ...-.1010I0412 IoIIIIdIIchIIoAIP IIAI o r.. =t_am.w u IH ”b.1111 w .m. ++¢:I.».I.oi. VHF.IIII?IA9 _o
Ur v It'li‘? - rOA .. p _ ‘OJI _n4yh I.. >. ‘ 9r*mo.utn.c_oww I06I9I007A ...F. _
«OoIooquboo. OIJIOIIL...._._ ...Ta....4«.HI ._.. “-9.1..- a..rI c L; Fifr I... v».. .A.
> I VIOIIOIOItyo no.0a 6.... Y+I¢ bDIfiAb w. I“ F U.» _IIA. 0- ll
10 . _IH » «17‘ co _ A! 1 .1
IL 9;...1‘0 1iIa‘IP...I r.+..‘ .WHAVHA JOIN: I4” .... ._ A.0_IYIW._ OI... 6TH .ava ..r
II 0...»? ..pvliln 0+ Jot o o . O 4.41r _Ib _ ,4._._ . I... - ,wII. ._ cc0.0¢_l+u
I: OOIOIIIIdIcIO§ H ......muc _vL YIIFLIIwfiIUIOAYHJHIH+fl+ .01 4.1.4 “L fit}-.. ...... .1.1II_-.I4I....
.... III._-L ..._..:-.....I.IH-H -IIA. I .I.. _ _m 7., I..-....-,.L.._._I..I.-._....T. J;
I] .HIHI- .. ILvIJ _ axura AH HH FrirArIf wH wpw IIIIin_r.IoIl.dfiW-.il ._.IIWIvIIl-ollultid.ml1.r._o
« 1 4 I . I! IIIIII. t,‘.. o 1 .
r31. LIIIHJIII...I .W. 74Ale Fur! - .“JHIWHAAw __1..H._vlfi- I 00.09I +._..:_..
IvIAA v... A fitchF‘OIk_l| .I.. < _.ch: .1 IH A Ff_ ’0‘. ...I ..i .DIO.OII.OIO - vc.¢‘_
ITAVL 0.... .1QLQtOoODHOI1lev ._rc ..ioYleIAAVIMIAIf. IA m-..l§n,'l..,fic4|1..LIbQ$. _._.Ir ._.
IIIVOQIOIOI*.1 IQontw». . . ‘IGIH (4% I; p H “.... Ll Io_..li._-fro.t-_o>| ~__~'u k
Irlvocco’IL I..! «.I. PTH.YQ to. p4 . *1 JJOIIOIQIo HUI *IIO choc I ..IOIQOIf
JIIvOflIOIOOIA I'oo.It.o+..PI¢ I ._.ovIvivuI I44 *LYOQ 0. 41+... ._iIIbo.or-+ 9.”...I..o_I.—.
({Or 09A ..J?‘1r.¢..0+ rdol'CthIbIIl Afivot A _o .L *Lrlhflfi l y. FIJI. out .4...
I 10.16 11 H w,wI...OL4A_ . A«. f _v .H 14.! Iv. 0+ o _ r41_t_l-_I. II f...» w .rIArLI.
. . . 4-..... . ._...I-ou 9 41.. II." 1.1..wa ._.-9-1.» I I...
0 :A Ifd..rlflloll.l._ VII" II. .IAwI JI.‘ . lrIILdL_+n..'VI.,..I.|c|II. WI uo~ o . _
“CO‘A L.... o -+.II 'IOIoI OIJrA ~M91... [0.1 __I “IrvL .I.! Ar. v A. a
yo. _ _# ....Q-r .IAJfoIIIoLrL .cv 4 HI . ._ ._ _ . .
_ ___ «I» Hr I IIvIo+Io~JIr.-._<.o ..
H... ._.~. I 99.0 0.. ..r... .LIfTrI I..-1 . 1....14-1, «1
I o . ..__I I .vIoIAVIo-WIOIOIOLI. a. .1. Ho ._-.I..I« rID COI'fd FIAVFOo + c. 0
fl} . 04 c c o .1 vtImoOua .._lu-..0I9.JLJ.1.I¢-k ..o ni 0
I v v 0.1; I._.I_ T...r.4. .IAI+I?oIo o 1' o q ’11,-.. ”I; .._. I A 74.. 0. ~. .1 -
o .. 1... .. ....OL. _. . .. .
WWW .JTLWHIOIII I.I-Ht... Ivvb— _.0443. I.I— OIHIHHUJ H¢FW~ 2*, H H
I H... “WQOIH Ll..l-L W?.AVDH0~ r...‘ H V IN.» NH %> I
p» H > P? r ? > . H I141 .1qu _
«A. _ A #14 A“ 96H H-LrLI 4A... III... o-~.¢ivolu .liOIOIIIIIPI 1
I .JHIOJL .1+.1._.-. ... . 4.1 » e44 .I .1a-+.+Io 7.1 “II .I.I.,.o.voc-
I .d .. ‘_ IIIoII IIQOwIJ‘ .. ..«9.“ 0911.6
1 .-.-7r . . .. . ..._,I.I...,ITA:I“T.;_ ._. ...A .-.-III-
+-@90|. *4. _.o- OQOOOtDIOI,..HII_.I1. 4I6_r.1 bfifiroo. afiIIAY..I IIQO
I. AYJVIfiIfiDIOL .._ c. 5070.11 .4- II. wow—__tx ITLIb‘n.ou- ._.. t
. I'L I,|I I A; I. o n. . o .O I .I4, I... s I Y O.
.I. . 3. 4_.I_._+- ._....ELIATf-I-Ir. (”1.3. H Ifi-I.
IRA-A age» .7ch n..., +.«..I.Ll ._. . . _...
4 _ ____H 4 a ._. PM II ,0! 4.0 ..so.» ..I‘IJ
1+, .WIIIh-I-IIJVL . A I.I I - 3.3:]...
_.L I413... -.ire .. I.I SJ . . I---.If.f.I;.I ...-«u.
H A... ._ « I-...Ilv... ....I. 0-1.+; ...II. 'IIA .- o o A on A.__,.u .I.!(q .
I o.Iv. ._o..,.,,. .._._.._.4V10Ica+.qo b: .9 l I .I.- _._ A._I Y's...
I I..? J....-H...-r..-r..-€.I.-wax”? .. L. _ 3-. I... :1 ._ II
I.. 1' I v _ v _ .. .. h«. _v _ J H o o whois‘ltwlfl O I o n .. $¢; I....If d 01v I..-.1; r ,_.I, .J. 6 .v O o u o- o h H - ~00. 1’ I) t
I W¢Y¢ 41.. I....Iufl +JHro+A .IonIoay...:IIv IAwI fiLTobHOIv: *...9oo~.s.vj. I.I..ITSIafibéI _I..I 9.7.9.- I ._ _
I w. .9. ..Hlv _+ ,+.I._...ro o... o. ..4¢.¢+I.H.T.¢1 ...-AT T1.I.P.vrA_.._. wII...fr-Io.t ...»: .... eo-AWoHVc
1 a 7.9 All . I .0. __ _ _ _ F . I} r» *1 I A. I.I.Ot L. Iro>vrpI It. III...OtIOI4I.. --O|OOoo u ._. ..Lv I. .
_ ._ . m .I. l... .. .4511... I. OIOIOJvI? .0 Do 4 .4 4 o _ . I . A. .I .4 I .I.”. I _.
u mwn , , ..Iqu. AAlIOhII...OH-IULP:H>IOO. A > Ll‘Imem.» «*{IIIHJMIHII 64¢II00 III. pm+>.l.._IIIHHHI-InfiJTW 7% WW
3 I.--13.... I-.-.57. . I. . .. r- . .31 run “I-I-... - ._._H-I:...I.--I I
I v *W. .I... I0 4 A .‘ .ov oOIV. .I....OOILAVIIIA 1* ‘uva01-Aml ..A.I”IOJILITAr.Is.IIOI‘Io IVIIIbiIOIAIH LL <I r.u0.l...9 o:- ..v. o . v 99
IIIOIIvlwfiM H... IOYQ‘ifiiTl‘2FAYYL‘:IYO Av! +~r7+vfoi+IITArl IIIOI... H.‘ 3710...; +1 . vOJOvH. _“.c_ :99
...-71.3- .... A. _ LIL; III... . A. 7:5. I 1:99.. L241: 21.; L I III...1TTL I p- ,
r ... . . A}. 7:12,: . I _ I .L...1..-L Li. Z {11. L ._.-15.42.23 .I . ...
§H+4I In MINI-”W IIITOIII'," » 1rA v H 1? d1$§.6*0 00%.”,a *LI #OO.O#L A1;+*.vu _.1IOII-OI+OAV ..VIOO ., .. _
. w 4... I 2. ..OOQIIOAAF_.vD._.AM >L >fl0000030 AI-.Iova W "9.4...‘L LTvOI4IOIu... ...:OOt. ‘lfll.IL on I a 9
f¢1m QHQIHOI+L AI .«.~4a‘ ' I fl—FrW 0s“. 50va “fiancv+ “I.I.teA T‘a4a 4 OIQOoIIouW H4» . _ .4IT .WIH «.4vVIwo_ v. 0.:- .
. a t - _ L . . I k .V .. Iv . .r; I I-II, 90 to v.0
n :2... .I.-$1 -I.-I. WLQ. .I . TL-If 7. ...-fill... .I.-I} 7 2 F 2m I11: QHIIHI P CiLIflhI. - _LZA . . H31:
_ _p 4 ... .<>o>pb < I . . 4 I|< 4 4
> «I. w P H #1 I 4 Nfi4 ._I. ... L=IDOIOH LI «*9 to. ca.+.v41IF ‘IPA “_Hv*.o. .rvLO INI QIIO V1,; 0M...
. L A ...-1.3 i- +:. - I-I-4 . 1 hr; . ... . ._.: M... ...-.....I...I ._....I +11 .I. ..
M NH 01909..“ .409IIcI . o... A .. I r 4 , l a . I.I «11.... ..__.. _ H ._. a I. W4 L ...LIIIrIrv AJI 9.?
4H IN.-vo9..., .I...&.OIO.0+0H4 ....Ncm? I..-..olfeoivll [VII-“.MAYJITHIQ.‘ -5400- u: w9++ O? .....fv4JoLToIfw-éa p. w...0v.H L. III-40.4 _ IAAIO +hu+v
, . 0 > V0 4 .1 w I I I I II Lr . I I . I?» r o I -II1 .1 or. +II 19.. _. . _ ..I I -
.0 no...-.L4i0.9.$ool_l+u-uwfi-H H~H TOIrJOtIA .I . 0.0.1 Inf-A. VLIbU¢+HImo_HJA.co_av4I Jrlw "ourxa ~r¢o0.,».¢.b.¢.6;I.IOIvc.aITO A .I. 1.0...9Mw
Ho. ooOI .‘ IOJIOIOIOoa. _. 4. H 4 fi II‘OH v. k . . .1R...n+ _14IAIrIOIIIdIHIrt IIOI #o-_.$1rAfiIOIr_4I.J ILLIWuooao. . HI. = YOI . [01 RT}
O 0.0... .H-OAIinIfOIbélol AILOv 0; o LI HIJHHI .. . .OIOIOIH.‘ ”HI-.I. to. >I 6Ho-000 .I14r13.0I0.»#~ c4cOJ‘A .1 1L {.I.-IbA 1'... 4
o 1909.?OI ill]... 4 1.6 b b 0 _oo : _t-TcI YA IUYI. HAT I I 1 L , . _YH4.I. o 00.. ...H. TM .u + _ o c v 4 m: I %I."uo_lb +fl , o _1 A a»; ‘ . t. 00 I .v . .I.-I.I'o I 1
V .HIooW-lo._vaxwnnchflwyu~HIHh1WA I I» OH ~.cL..O.I .I.".IvT I.. -Htre....fI_< . .wkmwv...H. ,._.V~.-HL_¢~_.01PW>T0> ,I» . v9$w M.“ m
o N P .- .i ...pII. c b . I _ b. Ir? 14p. 0 A r ..r 4 <4 1 N1 « H F — ..II HM 1H .1qu
I . HIIJII 1 1* I A 41 4 I I._I_ c- IHI’. VH .. o+L IfllloJ.IL_.IH. ..fi. . .‘o .00 .-I+I+r Av-.?_ . .. 9
I w.- 4HIA..H+A.~ _._ I... [4| ..OOIIWH. ...4M.qw.. “q H9 ¢A A I. o vohm m. .- I quo I..... ITIIHIOIIWIAVL),.IH..W...H VOIOIo. Q.._IV-9-Mo4-_c¢_fi4_,n ......" ‘v filo} 'IATITer L MtHH eH IAI¢A
. o, .dvmpoonu... .m4* . .r. :OIOALIL o. «....HLHIwVOIoU. IOAV I v A .IHY .ro . . I....o-ro 4 .0. ......«ar- V 5...... .... .... ~_ _. . acv+ r++loivo< II..._._I._.».. L‘. .0. o ...HLTI
1M0. In“... .wWI 0.”.FHWM: “.....Hu vuu“ .WAQlOHIHIOIuHO. » . ..qu-+I0.o Hvlrol‘.o.oo«Tc-II00.0IT_I+I,I.OI v ._vrI.oO|\_~I“1.A9: .OIO +vf+oo-.rc quIol.v I“ LT... a _."Aypm- o _.0.VIH-H..o. _v.OIHvWHHIH$ wncwfuauw HMHIMWIiAvHA
b t o o . . a I. a .. < , . n c o _ ,_ } ? a, h. r _ Ir .0! .1 . ..I . I I . _ . . s . o _ u _Il . Y... ...CI- . o . o a n c C . u I. .I.. O ‘4 _.
H.4+... P0. .0 n44.r0010¢ o .. - ¢ . c _. 10~+daaquOOJJIOI fi__I _ _Hlos... H Y OTW OI. o+IA v "_rolO OI”; ”(IH.UIH.HUI $.61 o .IhlpfuhlweIIIo HWQIOO . c. oI§¢ III-_d .7“ .0. .0. v r _ ‘ cw ‘. HI-do k “flufoo‘ctw AI. I1Jvh’mm... ._. ... 11H v
F .I ._.... ...u no . + 00% LI a4._oO-uooo.o.;..MA n-v . a to. o o. .Hc~¢ 1.. _OGC ” . r1: ._nI ,L. .,.o.u o o r~_. . .....toIOIO L.L.I oO.-.r.6 LIHIIJI. ,0: c¢.- v
iuHH. .9 H4W¢r IIOIOMOOL ,. +0 9.. u¢o_. .« ._.-tuoanuw A A 9 LA ._Iwr I c _. 1.IIIIO «103094 I OIOIL ,‘l_.'.s..*olv ltlr.Ill.oI I... +0 0.. "*0 1I Ir . O u H t... *Lflvf «.r ImI9rr orwr r
H,. o;”ofioo . ._.. I. k I a 4 cup. 0.. L7.Io.1uoO_TOpuOIooI6tmor .0 “h.“ coo-+oA .mqvaao. I’ I.IWIOIITIQIWIIIHIJHHHHL.VOI.IuOIml-HHWIAJ.9.WHIHI.HL.Hv a.” ._-A-.__ . .. Io. ._. u. ...IPI+IH1.O_I.O.IVFI¢WO%POHVO
-3 .. . .... - -- LI......-. I-I...i: 1.73,. ..-. L ...._.._ . ....A.~..§-III.__ 111+: , ,- __ A .. _ _ . I It - .
44. «um LI» H.v n. I... V'IIII_ I »-~ . I.., u ndp J» r»... 104 10. h>xvl91uviIlrLerl . f. Ah07‘¢.1 “Nu m 4
,voo..yH~.uI«. A IHIIH ..ucIAOLIr...4..ov .0.!9YII 5.... 9H4. '0. .o~_.r . . w __ ...... .4 {dumJTI ._IHI1- fi....¢1.o..ul VAIfiI ~0-:.-.I4IIH.HIL IL 1...- txf #WWLW
A woe+h$ou .llbtbru ..r-....--...... o... L. o o u~......v . .AOI0.0I.0II I) I.uo+ IwVI.1.uI.IO.mL.”a ___.IOIIleIv. .IIIWIOIIII IA ..1_.L p Yw
I ._- . no can . ...... .¢_...¢.. 0 I.. 17.1.4.4 'lwi-Iu 00100;..1.) . AIVUMO. v. 404‘. A . .4 “a 4 «I...- .. I] _ o .0 . J I. HI. v 1 {VFA
I o.».... - It..mu.ov Coo-Yin. aficfio I. 4...? 1'. I.IOIOIAo4IJaI $A. v. n ...“.«o .4 ._. J. _L . 1_IP.I..+YF
.IurI-01O. 0.. .H ... _uth . o «. .oW¢ 9‘ 04,... .I. II ...”. 4 .I .L _ QL+ viLIAAIQIIJ¢>
l ..OuomoIAIL!“ .I. .I..+I..Olabou . v7.07... I.I..Onv 00 o.’+»-*u # . .vooo om . b ..o OwiIOIvoau A
1 Qo¢ow.6>o.ouo.nu.oo . ._.—u.wco.. o._.«._.. . .aoa4o... r1.L.+IIHI.O»_w?o_LI To .o~¢o_vo3+4 r
_ _AI. 1 I I... _\ ,. 0.... but¢ «to. ... o... if." I. Lia-4.11. ..... ......m
I mud-2.3.3“ ..,”.I.”..;:.-..-I:.- ......zI- TIA-.. I“. - 7...}... .. . 0C5:-
> » up > {lit-OLI‘IIOI'IITI“ III I.I V4 1. 4<L4Lflqr J IOIIIWILTTLVWWWVOII
. r... .._I 1 .7... +9 hVW.IpIII+.Ovo,- ... I»..o.... .. . .” .....avoH- AIrLI fix—>1 76;»1
IA 7» O r ooa..oooow .019 OOvWOIfOO h...4.0cI .101 .t .I. .4 9.71.911.- HF II ._ .QIQIOIOAVI>w
V‘A Vfii f-‘AO toIO'I. : L..v..00;- otwoLw 00 4a.... . I... .0 4 I900... I004;- fi..IQIOIOI‘IO_..I-l «I
If u. r o..¢va-.;¢IoT ...-IL ....o-QI vnrc-«I'T .§ Io fivaoauoooo. moo...” .. «Iv ”WI <
. co ¢ 7.. .I....Ifoo? .05.... I.e. o._._.oI. . . ..~.h..a... .. ......«4LJIrfiII1
T» I.. a. c n 0‘... oocv.d-vv4OJI-Ib+t_+r1+or0 6A .- . o ...-0;. .o ....1 ...w».* a. vuVJ.
to 4. u ..w. c .. coo._.vc Loo-I.....TOdoIr-1H I— AVQI<+.9I,Y0.V4_1 OI»? A W...d+..ho. o. I
r» .. c9... ... ca...” . utew-vo-thw 4......0 I4- I- #w... o».,.. fidL<.HI4H. wr30v...mo*
I IVY... I- , ”A ..." A ._.” v . A?T_.._Y+II.I.AI 1. .I.-.Ih.+I..o.v ...v|_. ‘nitujj »
I 9 filo-ffraoofl .4 0%.... bléfquabL'loo I.. > 0 1T0 0.00;? ...! .I +I...I+o&..HI,
I.. J. . ...-,_.+ TI... -I....H....I:__;.... ._T . .i;...._1* I;
I . .mo 00-- 17.19.. .I. n O .1¢oo_ooo.. ”.... A_.aoo _g .Io . .._._.t re_o I
In ... A. u... . I1 ._oavooo ._.-._...TII‘IQI T»? ... .~ - o. _H . ”ML. ro. .. . .ro.¢
“T; A... _ 09- o A OI'JOIQIOIQIVIATM'00"00 ... t o... 4..“ ¢ _H Y+lIrw-‘+ .‘Fomrbs‘AI
I t o. . goo; . co 4.4 O.AY.I40.MOOJ‘O .{wa IAI o .I.. .. ... _ . _ . L4Ir10.. Iva?v.u 9.1
In M.” 00-11;.J-mIW4.790 VAI thug... +a-v .- ..¢ ..._ on: a a ...!fI— oI A... flhwh. . _. u. __;L vNQfiu+A4H1wIID #1
II Arc. cc 9 .44_9.IcaI10.; I ..I OIIIY-.:,_..opo . AA. .. a... .... _a. Y .. 1 IA e.._e vo-Oov.‘l
.PO 0 . m.... __ ......41...i . _ ooLI._I»I .I»v_ n 01.... ..p ..L IL *wrv; ...9.~ .
Iv; w. .0. .t.I.oOH-. . .n. ”AIvv 0...,IYH0 I4 _rI I.I u.u. , ._.-II. _ “<Hh4ww ”-
» bun-._. - pr» s» > . 4.4.! 4 ' IIs‘o.'.I.vOIIA III
T C... .43.. .4 TI .0.-43L ..I I.I“; fl 3TH ”$1.... . .. :1 W I.I... 4W
, - . It ... . I I.I. .I‘IIY v .c o o 9 . a . u . . a n - U I O I IIII.II I L75
u ”M .a lcwI-IIWJI*_QIHHA1I_O v.“"..- . 4A-a_u. 4 1.... .am. 0101.. LII.- 01 In . .- ...v - .o“ ”.11 LN
00 _-o 101. _._fiT-u .9 III 00-...4 _.JI._ .I.. ..IO,0¢..I-OOIOrOoPIIooIo_ .41IAVL 04 you . u. I... 0... «994'
“Q. 4I¢¢I¢Q-¢v4M-_100. 0.0 -0 006+...» I4....OI¢. .I.-I..; 00.0.5.0.'b|o Y"! 00.5.6 OJJII .7“. 0‘ ‘Ir . . I6. 0H ..114
1- -1- .- .:.AI ..TIII...._I_I_?T 1;.I_..._.._..I....;. ._....t... .. . A a... . T11
0. ...-...ul.._.a 0”. Iv... HTtIvifoIo .quoé ......4Itoo .. ..v .4IJ_IoIa1 {.IOIYI -; a. u .0 .. ._V vkwooosAAXyI
M ___ o. .I.-09* v cc‘a a... “co. ... .ao._.o‘ ‘_o~.. cw 1009.?401a _. _Hu ._.- ¢ . Ir. \IbIr.VI.v
“9*. >. ...-at I+ .9... a... .H... w Acofict 1... __o_ no... _moufllHIHI‘IYO IzleIoI 01a...“ a. ...tIIIAP. Ill .Lnb _.L _
'0... ‘J 11414 (4‘10... CIA-1H L“ L 0 1.4 0 m<40ainoJAfiificIflll _ OQIuJIA1H' .0..L LI I..-I“... TTOLr
r. I Jo‘ ILtItIo ..I- 0.‘IIQI04-Oo *OI +9GOIOI¢A +IOOI9+0¢r4 0 00.” _.1-1 A q _ nc_ _coL..U I. I .5 4 A
. .\.\I'4 _ A. . r ......O;¥U.OOO. . v. Ihco§r O r. I a —1 .0. _I fi
olsI..IlI,- ... . ov 00o¢001<x T4104<00 .044. ___ _._ ..I - .I.-_TIQIIQO t. w.. ....u cova. ¢ocw.. “IAA L
- co. ....wHP+Wo I t *. J?.,.’Il0( [OI-“IOOIATYOIOALIYII 4A<1<a o ...”.I.FI1_I _u. v.. ... _ o... F u .. 4- I.’ ... .Qv rAfi I
v.1.4 f v.05; .OTIQ 45+. .0.... r «..mLcr.....¢ ._,.ca..oo.vov. "4.)._ .. 9._.o ...IIIOI..LT..II I... — «cc-v4:
{_rooo 4.H.c+. . o. ¢+.9I4I?I!HI.I. 5'91. 1 H 441.740—‘3‘4... .. o—._o~ .. ”I .I n... I T“; I ,-.. .— _.... o ..«M H m
01.4 If IO To i1_._o. v 00.5.... a.. _ .a.g.-.c- q o c h _‘_ o..I o. o
. , In r¢.ua.. g.. . a... ....v .I ¢¢..a.a,..+ . .I.: I. II.II _ . _._ ._ . an ~—
4 _ _ . F .. . . .. .10 .. A. so I an
- ~._.‘4__._ he.” aw... ..Ib _u4 f904o. I h r ._.... ._oI- _._ H. i.. .A
A _ 2 [VIII. IIILCIL. \.I . ...
.. :..T_L LL3L2fiEI HIT-um. w. I.In_I_ II _IL . ll... .

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

DEPARTM ENT OF MATHEMATIC.

MICHIGAN STATE COLLEGE

’09. two-

fnu' ‘OnOUI

I. H... .I.]nlurc?ollr.t|
_
.6 1, i

J :.—.I.

|_.}.\..r , .
00 04vL|I|¢|-|'4I. +
.. u o < v .o .. _ H

-
50...... neoclo‘.v.1.a.n|._ .
In...¢-no .Ao '7\¢¢...l¢+.1.0o.
'OIIII'OI'IIIJIIIO-Ottfiuffl'lur -.- nul'lln...n a.
._ . . ‘ . H. ...x. ?
‘ Q.

‘illfillll 0L 4|. rd}.
I.I. o -.l. ‘1»:.ll.lI..

_.
.0
v
_r
v
.>
_
r

w u . . u o a “n p o ..
cliALliOl. $.31. yIfO-O 1.
_V. c?........ 7!
ydlylo . ’r 11"... l I IAIO:
, - .- . a, O O
.L. .llOt V
, ”1.4
,O..,\.TA. iv, ,, .41. I - I.I: A
I 'l“-.+nlll'.bl' QSH
. r MILLIY?L.[.1...§.,"r ”\fillul '0 I‘vlfllf ll"|r.t.l~.l.1..fi:;. «
.... ... 1671. 5.1+. y... o.¢.-,-9.o.._i. m‘Y...
.....IOIo.9_... o LoLm .. o [01"1101 0.947- n O.
_ _Iobro’o ’a.’ 0.III|. I‘ ..Vo__I I701".
O 'LOIQOOV96Ié

0..

¢

.

o I.....Jfo.
¢_¢o§¢.v;1+~¢
+F‘ 9.0.7006‘6 ’:I..QI¢

+09144J+4II+
... 9c+.>. a;

+.ovov¢

V

vh*.r'.O-L 9“

...Yl ' ‘ ill ’5.

a o.
.Oun,o.IO_F.u. ...
n.0'. _

. _ . __ _ . .
w LwaltLPLuLlL-lllLlnlh

 

DEPARTM ENT OF MATHEMATIC.

MICHIGAN STATE COLLEGE

 

 

 

 

        
  

                        
       
      

 

 

 
 

 

 

 

 

 

 

 

 

 

 

      

 
 

 

 

 

 

 

 

 

.i‘Ill‘l|I II. 4 1 I.I. H1 . _‘III III! I.I- II I III. I. III II'I'I I IIIIIII'IIII‘IIIIJ
II... I...I...I _ I_i .. . I . IIII I.. I . I I I. I .A I.. II I II .
_ 1 . .. _ . _ _._ .
I10:- I I.. I I .I.: I... ..II. .0. _ | .I _ _ I . I I ._. I I
III 0. I II II Inn“ _ 1 OI _Il . _ _I I _ _ r
I I I 1 .I I I_ _I .I. II. I If I. ..II I III I _ I I. I I I I
I0. loud I a. .... v I— II I. II II. III4 I.I! V«_ . _ I d . _ I _I _ _ I
1.9 I. _ I I. II. b _ __I aw I.....r _ It caug IIV I _ o a! . ‘ _ I IIJ I I _
I IIQQ I»- ..I . . I. ..II ... I . uIIOhIIOJOIQI I.I . _ .I . III _..I I I I
VIII. II -I.. .... _,#.. J II.I.I I I. . I. ..II c ..hIIIOIIIoIIIJ. I no. I._ II . _. II. I . II. I _ I
'96. _ _I_IoL. III. .I- I...— ._IYL_\ .1 I.I.III 1-7,. I n .n .I no I.II _- I I. II J I. I.I , I I I I I _
IIII14 11II1I III I1 J..,.. I ‘41: .l'. I I I.I...bIIIIIIO I.I.Jul. ._. TI+IIJIIII D .. I I _._II._ _ I III I.. I
II I ._.II I II1 . h . ... I I.IL I I II IIIIIIIIIIIIIIL. 0+ OI. . I 9. II I I V . I. I ._ I I .
I 0 II_VI. _ o _— a. II III 9. .I....IODI.IIOVOO-t I I. I I . .. . w I.
Y IIIIIIYUIVIIV I. aoLIJIIII _ ...IIII.JII ..II: I. .I u I _ .‘I I I
III. 4 III In I _IFII o . __ I I.IIO!IIIIIII __ I II, I II I .III _
fio . o o. I... LL... ..II. . ... ..II ..II.I..I.v.III I I . I I P
I..... I 1 ‘ II ‘ I‘. _ _ ..II! IIIIYIIIIIIHIIIA I I IIJI _;
I, I I.I.I . ..II: III I. I I I.“ I
I _ I 8
.I. .I I
_ I H .5 I
.I _ H .. .LII I d ..
..III ..II M __ II I . I I. .. _ II II.
. ..I I. . I _
5 .. I II I
I. _ I . I I . .U I. I _ I
o _._ I I I IL I I I.
4 I . _
II I _ _ I I. I I.
I ' I 1 I II I III
I. _ I. I... I I . I. _ . I I
_ ... .I _ I I” _ I_ I .
__ . _ _ I
I _ v I I I I
I I. ‘ I .
I II P I n _ u
A _ _ ..II. II I I I
_ _
M. H u _ Tun-I.I. _ I .
_ I I; II _ I” I II. I _ I _
I.I I II._ . _ I II I _ .Iv I a
.II III IIIII III. I _ I.I I
~ 1. ._._\|._.I .I.. .I I._. I. I l _ _ m
_.o «II'I (1.1 III I.IIwI I H I . .
. . I I I I . I
I l _ l
. I _ I I _
r I
u ,I I I v I: I f. _ I,
.I. I OlY ..II ..
._. I II I _ I
CI I I I! .
I __ _ I g I I |
I I I I _._" II I
C U I.. II _
. . _ _ . V
. . \I\.. I I I _
I. I I: I
O .. II. I l III II II I ‘ I. . I I
O . ._ I I I I III 9 I I I I. _ __ _. I I ._ I.
._ I - . I I I . I I.I IIII I III _ L I I I
II II... v.0 II I I.. III I __ u ._.I I _ I I I .I II I l
IQIIIII II.._. «IH _._1. v. I.. ___oIII _ . _ 1.. _ _I I I I -
I I I. I adult VII 9.. .I., J. I n r _. I II _ I III. _ I . I I I
I.IIIIIOII It _.1 . II I I I I I
a u.......... I ._"I . _ . I I. I.
.11ij 3.”. ”no 9| Ifi 1+1 fl _ _ I I. . _ . . II
I IIII_.I. _. v ._. I I . O ...__ ..I I III_ I..:
II. II .9 611.0... L1. __ I ..II _ I I _ _
I.I «IItI.JIcIV «MIMI. L. In I _
IIIIII .IQI* ..I I. . : I I I A II . I
_f. I.. 1r I I... I .
.I .I. “I I I II4.‘ ___. I _I II I I. I I
II I _. _ III .I. II I I
It‘llI_ _ ...vI. vv‘ I . _I .. 1
.I.- O_o. _ 1 away 0. I I I I IL. I _ I
IL, _. on . , .“IVOIOI u I I . I _ . .
.I ,I____I I.IIIIIIIIIIVIIIIITH I I I.. III. _ II _. u
ItUIIan IRIIIIIIIIPIH'III :IIiIIIfl-IIII IJI . IIII.II_ I. I I . I.
. .11 ..IM_ .... 1‘ 0. ......w,... Ila". I I.. _ II_I.II IVIIII I . I I
II¢+I O ConclavwvnIt.‘ co . III. I I _
I II ._.aIr I. I III IfiIIIWL III I I I v I.
I-I.¢... III. I a.. 0. ...tiv _ I I
II I 0' I.I..I I+_IY.._.flIIVIC | I I .I..
I:. I I u o. I”. I.b _ I ._
I_ .o.« . I.. I_I_I.I—I.I.I .. ._ I I
III I” IIII..I I.I“.I II. I I . LII- __
I.__.._I..v.I o _a:9IIvm.U.o I II_ I. I» II.
.91.. II... .IUIII.1.flfOIUIrI.I‘IO I 5..IIFIIII I: II I I II
_ImI .I.. I ..II... .... I, I A I I III II.
_. .I ._1- I”. .4I . I _ I ._ I, II
.I. Iv... ..II. I) IIOIT-I . I I II I ._
_NoIIrvIO<IIO.o.I._IIII1 I o I. .. _ h __ I I
I gchVI‘. ..II. I". OIfiIOIIHJO‘ I. I I _ II o . I . _ _
a I _ .o . ..11... o.._a._oo ___ _
4 ' .QII I I04‘_I.II I _._owIIIII_I.¢I I _ I. _ .
vII. w... .I. I II 1 . I IIOIInaac IIOVOQIIIIII troI'IooIIOIoL ._I ...-I II . I I II III.
'I.II._III _._ 1 I I _I__ I.I.III._,Iao.oIobII_ObIoIIIII_ IcofiIIII._I 1 I _ I I .I I ._L
I 01004! I .0. .l..O_ I .fi .. {Irvine-{J III-I.IGIOIII'o-(OIILI'H III-I. I .I I . ._
I..... “......IIIII JIJAD. i .I.... 0<. ..fitu.n__r . I I I I
..II... I.I I II I _ I... I. t. I. ...IIWIQo..I o Hfi‘ I. I. I _ II_I I I I I II I
v.1- .- I... ....II. _._rI.no I.- I .IIIIIO IO¢“‘__I_I_I.IOII _ _ _ I
I In. .1 nah ., p9.v4 III... ._.... .I....OronOoII . I _ . I.
L. I I. I III...Io-..III ...-._...... I..._I_.«I- I . I . I
T ._ I1IHIIIIIOV oIIIIIvflI.ooIOIIvo ..II ..I to ._. I II
I I I I.. .0... . _I. r I. .I.. I I 9, I.. II I I I. I r
QIIIJII .11 I I. ..II. I I ..ItI “tIVIIQIfO VIII! 1.. I. ...- ... I v I _ .-
fiII . I .. I_ I I. ”I ..I... II, $3.9II1III o Ir._.oI . IIIJAIII... I... . .I I . -
1._OI1« I IIIIIOIIIIOHO A++I IIOOII ”Ii-ar_1¢...oo 9. I.. I II I... _r II_I
v. t I d IJI_I.Im‘ . TIT (IflOII ..Q. 5.. _DI. I I. I _ _ _
v. I .I ..II”....”: I.IIII ..II. .I fl IVIOV+ .I. ._. ._ .. . I I."
roLo I.I-IIIIII IrI I.1. II. III II 1_. II It I ___. I
vIIVII . .I. 1 I .I.-.... II IVI1III+¢Iv I. o _ .I.. _. I .
I..... .m..... _ In. ,I ..u I... v¢.—._I_1mu I I o v I.. u .
v0.7. .. II.._I. I o IOIIOII IIILIItIv ..II I.__. .I_ I .I It _I . I
v. 0 VI- . I ._ o I I. I O. I_I L.- . I.I ._. .I. I U. I II I ._
I? . .... I ._ . . ___ .. ......JI. . I «. ...... . ._ . ._. .
I II II I . I . ... ._. I .. I
“243. u .7..3.f1w .I....* .3”: .LT. u. ._.. .7 ,. T _ . L
L .. ...Imh1mH4III.- IJij_ _FOIWO '—I.+ +I1I~+JII HIMWI “Hi.“ _—I . . .1 ..I— II +—.&H|U|:
vOOOKIII 9. I.. II II I II I II I._vo,.oII0J _I r I II ..II I I_ _ . .. _.
VIbII'IO .I_I. OI _.II I I..-o _ o r_. . __II I ._I I _ ._.—
00 .1... I. 1r I I .. ... I n I 0 II. a _ _ I
... c ... I. I .I. a I .1 _A
OI I _ o I In; _ I. I.. v r.
... . 0 .I II I1 _ h _ .-
19.9.4.3 I. ...;p. I u I . I. u _ _ _ _ I L
I. u: u. I I .I. I F. .Ir u _I-
I'IH _ II '9 II I I I I
x. .I J I. I. ..I II
I... I . I I. I I..!.. _
II._I I». I II I III V
O; I I I .0 . ._.. _
I‘IFI. ..II I I. v 0.... .
‘hIIIa . LI .I .I _ _Wu 0.
It_ I". .I. .. I IIfII I
IAIIIIO I I I.I _III. II I
#- ¢«_.I.II __ __ _ I --..I...II. I
Ir.itfti|.i.Ltil'-r I II _ I ..II

 

 

 

 

 

In} II.I-Ni \ HAITI-1| -U".'II\;,I|

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

     

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

 

 

  

 

 

 

 

         

 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

DEPARTM ENT OF MATHEMAT'ICI

 

 

 

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

    
   

 

 

 

  

 

 

 

 

MICHIGAN STATE COLLEGE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

 

  

 

     

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

H4HHHH .quoiqu “HIM-H“. I
... H “4 0.. ...A. ...4._I.I._ .44. .H- 4 a . 4. H HI. . ..ouu H
O. o I o. ..I ... I. .I....O... , . . . IHF

I 4L4 o.-..+. I. II. 0.. . I. . .. I..... I o I . ..IIIJ; HM IH

000.. ...... _Io..lw .. I ..n . If..l .I. _u- 4.4

v A 00 o‘vl‘IiIilcofi I. 0.4 06¢ .I .... 1... . '3..-.. ....O «7
LOIYOOI._OO4.3_I. 0 .LI I .o I.. .. , _ “I..... .4.
100-..... ....443. .I... I I. .I o I.4IA I4. I .I I ..va9
V¢ooooowO %.u.¢_lt..-.. 3... .o I mo.4¢4..oII49II .Io- +. .Y I I_ ._.. IWQHHHLI Y
4 4 4 9.. ...?J {4 . I _ 4 _I . I
vooOIoOofihdHI..—_p. I III- . Hqfi:._oo{oo _Iulr‘li ...JafilIVUQ’IO “I.“ ..4 OcanwA va A
00 IQIYQOIIaIa ....n III-OI.-. fit”. o-IIQ. 0|,o-II_I... 4 ..I, I¢ )1 I..44I. .- 4
76-0 "IOIrIcIaIH _ n . 4.4344 .0.?cI,0..C1¢._ v ... “.44... omJfio .444:

Irv... ..."... olv 0.: ..ur ..M 0-7... flIrIWUII .. .- .444...IIT.. +41... 4940+;

f‘o“o.o.vo¢94n._uo 1.9.0....o.1. 4444 y. ..w. PVIH... 4. I. 4,. . . .0‘Vwoorf«o

f...on.n..l.-O|O II I. ..F I __ r 4.3.... I _ WI..F..I04.I _._... _..1II .I.. ....QI..I..d.vQ-9L.Io

Y‘AOOO'.IIOI'._..IAIvILIfi o .. .0 4u4fi ..rKVOL.._.OrI4 '0. ..II“ I _._ _ C t. TL O+ 30.. df+olIOIIO>Q

I]TOO...-Iov..o« ... .. ...-.1 4...... .I..1.4.dw._u..: I..... _. a 1-0.. +.._I¢-_a....ho.oloooI.1

IILOtI IOIOI.) I.. I. I . .r. III-krl . 4. rY‘44 .A .. 4.. t-Iov IF I... ...l'.-.I4 II .I. II» '1‘... .III CIl...‘

I» . . 4. I ...-‘I:|O.II .I..00I.. .0. I 03 _ . .r 1.... to . . I.. _ . . .-
«F rIl-IL.!. LIA-Flu!!! 4T6... .w..» 4 . 4 ...-.... I.. 4.9 v Ooeo.. II ..
Q00900+¢‘I.I?U.0-.+ I 4. ..4. 4N1;4I I.. I I..-.4. YI-fillIIIO I...- ..‘JI
Ufioo.co.0 ¢.§L_o .vv . .0 L ..4.. _. I.. I aIrIIrIlob. .. 4 I II'- nit-III ..II I.I-I-I.iqluliitlll
4 4 ca. 0.... Io-o.vIou-Io.6! It .. . .
’50....00a94 I [l-III.I.0 o FVLIV .4 4 . I.. ... . . . .

v.0000o'l-/ ..IIH'OOIk.O to. #04. a14-IL . 4 ,.

1.6-04.0... .0.?04- o-.. I.. . ..- 44 H. 1...... .

IYAY‘OOHO YII.QI10F4I40I.H IL.I o ~.Hr ww. . I A 0.HFII -.I' 1

II 9“ ailmfiuv*vt I _ Ft... 4. w”... r4. I t. .4. I. I H

Y: 6 to. I 0......IIII I-OIOIOIUOI . : .4 .... 4 4. . l. ,

II outoao+ouf+o v ,. ... wow” w4 no.4 4 Me-oI....q..

.n40P44Frol‘L 41.....(IIIO-“dl6 IF I.I . 4 w; "4. .O I
II OVIHH.“IFI¢0 ...-.447. cog-o. k. . I”. . .0. .. ..u..I ‘tII.I. ‘0
11 4-0 0. 1.10 p .1. w I 4 c o o ‘ .é I.. I. . o I I. 4 414-??? .Hi-i 1-4. I I.“

II Ono...Ub.H. 4.. wuoo .0.— .4... I... [... .I.-II .

_ . 4 4 I .- I o IIO-I .... 4 ..3. I; o. o 14 .

II A _ I .r. I . +4 . wt I GI”. .l..|.[.. .. 0 0.0. O . I I . .. 4 ....l ....[

It‘s paah‘o4vhlv 1.4..bvv-ooua Fry... 9 I 4.. _.IIIT_.I .w

11.0»-.9...H.ov..o._ 1 ... n O... Y. I. 4 ._. 4 Id ..I.

TIL n WO»4 v.‘ .t. .00.... O ..4 .. I I V 4 » u o . 4. I a... . c I

. ... . . . . _ . . .V
HI 4* o .04... .3|.. 4 0 _ I .II .4. o _ .- .. I
H» 0 . .... o .. . . _ I.I.-. o.
.— 4 P . ._ ..
I: H H. J...1 14 . 4 I.. .
VAvA ”. +.IOIO. lfr'biIb 4 l. I _ I _II .F.Ii.l: :-
r. .I.... O I. ...\ [I.. .... r . .... ..

v‘3LH H _ _ 10.4 (I

. . ¢A+ru91 .I ..o I.. I... .0 .
v5.4. Q I. 0 O. . _ 1.l.4. _

...; _- ....._.-.-4 4..

U. r.. 4 .. 4.. II ItLI... . ”Eh! C.L
..I. I... o ..
II... . «.I I {III
.........I ..II. I O
. 4 4 4-..-.I. I .o
. II I .I.-I :- ._.l I I .I..
o .1 . 4. 4 . . .
o .4 . I I ..I4 . I .I.. . . .
C .Fu4 4 O ._.4. ill .
b I .4 . I. I4.
.II I I... ..II .»
N.A . .. _ ._r.l...r I ..4. _‘O"“
.0. . .4.. .31..- d. .I4IY‘V. I.. .4 o I. ‘ .4
Db»? It I ._.L ..II: ... III? too-o4..w..4...s A
I 4 4... . 4 I _. . .I -on I.I.rr.

o 4 . . .4. . J .-0
E0.”- «h4v.I¢-r-H ”I _._ 1 [0.1.1. 4.0 4 I744...” .1 I0
10010-0 I.I. VH0 4. o .4 b4 4 4. _ ~ 4. _ .. .. . I.I:IHIM.¥A . ..
I........ o. .... u 4.. 4.....4IIILIv;h. I..... .+. 4.... ..fl..++ c. ... .3
100 12.0.4 . I. I. ...L- ._ .... I .. III. I0... .. ... .. . ..1 “Ind. 4.. 947
3. .... .. .. .. I.....I. _f...-......__..._.._._._..;...._... .
I..... ... .... _ 4. .. .4. .4. 4...-4.4. I. .I.... 4...-.14.-..w.+.oI.o+4...?r44L.r
r900... I.I . .. 0.. if... gr _4 . .oo4r. Ocovr 0 IOTIILI IIJA.’.r.I..I.I.IIII.-L. > »
Ivav-. . .. .41. o....1 I.» 30 ‘44,.1_ 00090....0I I’AO(|I0'1I.If .I.-4.0.191... I4..I. I .D

L ...&.¢..tf+.‘4yi QI+ . v.40900. AI!‘IQO¢I.OF|4I3 ol'vnon _.I ..l O . 10". F 0;]

H 04.1.4.1- ..- . ..+1..-.I_-I.-I._.I..... I

4 M w“ .. ._ oo- .. _. II I..-..Olfl. I'D-11.1’4TOIA.‘ I»
H _. 4w. 4.4;. w 4. I. .. , F . _. L.
w . ... . 4. . Hok o...4. o¢04~4o4w¢l A
4‘4 . .....10110 5114.7»...6... “:9... 9. ad. .... A .0 4 . I.. e. .1 o «I. «.4.... I we 4, 4 3 .40

11.94.... _IHn4 4..-WI... 610* I... I+0+I4IW4 (v.0 I. 1.4.4.4..” 4.4....rua. ......o4.a.r.

..II. ... . r i .. -4 .I--.-.-.. ---- .I.. ._.... Y 4.... 4.. 1.1.4.1474... 1

XIV-..Iro r+I+v1wvogo ......o.‘ 0+4-vcooco or owe. , “.4.-II 4..__~ 016.0... ......c A I

117.. ..v. 4... ..qu-4_1 04.04I#4... ..ITo.II-¢o-..+.: .1... ....o§ o.- . o~o¢00.' 4..» .. 4 LI

-. ...I.. 0.40 4 ...:I. vVLY. 9.99?‘ d.. ...vwlfx .. u . Ca air. I-..J.b-,lfi|v.0btbo 4+...av9 LI

“my. ..WOHII‘II’A 4+§4+.WHIVAY<¢OI. .4~O+ W... ”0 .k.l.9...:|.-..I..Inr «I. ._.-0 Oil} 0 .I....I 4 I.Ifl 4...-0.0.1] I.I...O.l
1‘4 4 4 1 . ... . ....4. . . .

UVHQHI“ .... .4.... u... .....4F<.... (0‘ __ .— fifi. N ‘1. 4MP. 16*..." I.. ,4.‘90I000H LNOHII A1
to... I....&.....3ofinroo-.-I ,4. _1~—_4o‘ 4 HM I.I.AI_IflIHA.\IIII.4-NIHHHVH.40Jt .- On... 44 fl «4 AAA:
30“. ‘00...o.'a.&0 . quNOAQvatn new. .. ...- I I, VITO-.40. I..-v .IIOIO.I- «Ir-kl 9 I... O. Oo..o4 ALIAI
vet I c.¢4..V44....4 4....4.o4v4 74.03.444 >4 .6. f 4 o tfa. I .I. I. I I. v Ivaono- O...-

iov44.. ._ To...o+Iv..o.4...co .. . ...r..o4.oc...d.¢..?_¢4¢ I.. «at .... ....Loc44 44.4 T .. _ _. . . 4

W. .4 H vOootor._’I c.0990... 1.. .flrY’VHIQIO ...e. ..4 .00» o 4o '19 . o .t,

I".~HI ._ I. .... .I. . «.I.... .4: _ .. r .” ....

I .t a . .r 0 .v r y 4 ‘4 IrI 4.. I _. c

J£¢£I.OI..II o}. .... .. h . . L ..n .

v HIooHOO.OIJ.I .fi.4..m F. o b 4 .1 If. I

Iv (tit. .OIQImd. so. I... . I I-II4PII I O.

1.00.... .o I..... TI _ . III. . ..

IV. .I.. ‘4... 0 VIII .4
PHD-40 0000. o.‘ I.. O 0.. _._._

I IOOV IO.OI.OOI ..II n “.60

u. . I I.I

I 0 II _. . 3 o I t..- a I; I.

u .... rIUI. ..Ioowv. N .0 .1 Y. o

3.. . ..I I.I... .I.. o Oflflo.OIo--Io I‘VQILIIIQIL. . II II ' I . t .5 00' «o d

.~.n...III.I.l..J1I. wrlfI'Ir 4i.» » .4 F44 . 41%| T ... J _ two»; 4 I. ...44 4 m* I.I...vo. 4 I .
"1 .. .A_IQ-69. ..II.“irb..r..rlo-MQH.‘01...QIOII4O. Ir III +«HL '49 I I . 'lf. 4. W1 4

3 I I. . I.. ...-I. 0.34 o. .‘.I I 0H3 . +4aoec+fiI> {To 4.._.I4I9.4 ..Io 44 I AIIII-I .....I 114”.

Y4 .14... ..II. .4.r009 04:390. 00.. .‘.009 0000- .4.-...... 4.0 .I.-I.. a I _wdflo 41- I.I..4.1 . . . I

I I v .0. I... cocoonchvI 0¢.o.sl It..- 1...; r0... _._ ..II 9. .4. o o .r I. v 1.. +I o I 4

n .. . v- 0... vl-Q-OI .tIItII I wilt-Jr... .. ”II-OI trov . 90 _ .o. , I . o .I.; 1...

.... U ‘..I¢ #4 6.. I. o I I I _ 01 I

I . 60.. .4 o I a I.I“

I I .

I I —

I ..

v -. 3.

.I I4 ..'d.l- I
.o .I I ..“.o
filr.-_I 3.4 I...
. .I .4 ...
“.0 0 OH O _
.. .. i a .
_I 4r. ”.4 ... o a a «It-IIII 444
I r i . I I O .o Ivu v. I to“ . I .4
M4” ”I.“ “ .l I... '9'... 3a... W+ A

u............. - 1.4.. 4 .-I: .734: ..

...-6.1 o. o. - O4 ou44um..WH 41!.

Yd I o ... ..vor4L .I..»

If. ...... . . 4+44I +4....

I..-[0.6. 4.4 . a 4 ._awV

re... .14. w. ._....vo «_.a_ 4m.aa~v.o.w41

I..... 4... . I .. f. I .... .- I.I.vl

I... I _._. . . . I r . .0.-I Hy o..«4..o cmfifl+9 41

“I..-47...? _ _ I . o .4 .. v.....0§I.9I4._._o*. o. 9 AT
III .. .I1.... I I .. . . .I. 4 .I _.n 4. . v.._. . HLWIVHCA I
Ilfia. Law4ra . u I. I+ I. II. 00 '1 f H4 '0 . .4 0.414L44 01
II .0. 04.1-. _ . . .. . ., .4 . .I. 4.. . .... ..
Ilroo .4..4.u.1 . r. .u I.. ... . ....n.r.w.r.v

I I .. I o _ .04. 3.4.. .

‘ITT . A 4. , I4 ._ . 3 I...- ..1 «44. _.‘wvvf

p .-.-4. ... . . I ...I. . T. L

n... . L.. . . . _ ... _ 4. . ._.:4 ...... +4...fi++nflo

RIO-fr. .... 1 4 I . no I .4 I19... +34. noctuth-IHOWIwImeQ
Y Wnn—o 1....-. o o ._ 4,. .3 - 0000.04O.OIIoOI o I. ..- ..‘ ..

. II...u.. _ I .4I4I44o . uoc,—w4a... It. ”I.....oon "H o
I .. . .
I.. F? b . 4- I .h .4 b _ I H Ico*N9.O-Q~IOH “-

 

 

 

 

 

 

 

MICHIGAN STATE COLLEGE

 

 

 

 

 

                 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

   

 

 

 

       

 

 

 

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

      

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   
      

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   
      

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4“‘1J‘ 1“
r V II .IIHII 7. .
111 1 411 11 4144‘ 4 «I H 1 <1 of; 1“. a .IOVQ§
HHHIItIOIDIH 001* HO ..II .I. H ~4wu“ {.I.-I... IFII J H vIle 0H0. or . .I. I II‘
«I I ...II. I. .‘ II... I. I I. 1 fl. I7wd4q “ .Iovvn..“ H OIJIOIIIIQIIIOI? V . . .r t O.VII'I...II .. I III—I I
I u» I .I I .I I I. 1.. I .. ..». . I1 :9 I Y!“ . ¢I?0I0.rlio . onOOIIIoo. I ...?IIaIII YQII
V «I. LIIOL {1 I'QIanllo... _I.IVII- ..II -I In- ...I1I70.I.I. ..II. TNT” I. ‘ . LL . o . . o a . I . .10. II. VIII} .... flQA I. .0. .m I.
IvIIQTTIJ I..?... II YIIIIVI, III a II Ir I I.L .IIOIfiIH JvIWIulal 4| .1 r. '0 OIII?I _ II! [II 1. .I IIOII‘IO.QQI140
1 A .ono... I I I. I I.. .4. I I.HH »o *1 rfirlfmoa. .9; > . .I. I09.» IL..I.I.IT9LI~IIII.IIIOO.
I II . 1! .I' IF. IIQII [I.II L .1 III III VIII..11_ 1 .1 .. I _I w . ... 3,91,. _I _I IGrP'Iv: “_.TI..OI.IIOII. V. II
.I. I O: ._. . r e. r . ....... o I An. 40:99.39 .. fl
”I‘D.“ .OIIII [..IOII IC.Y .4.. ..I... I...I‘..VI.II.‘ filo‘ I .I I I.IL ¢ w I1 I .09 QOI I! .L . 4
vAI III”....V.I H... . w....o.~.. ...-AA .«m ._r I— I—_ _. H lrliiw
111¢1011L5lolu .0 .IQQ‘IIIIII 1.10. 1.1]. .&_Iw.. A .IN .I. A :9... 1|“ o (I . r+r<+I_.4.I
: ..II.I.. ..II. t.. .. .. c o. IJaQYoIIo.+_. .H _II? III .I_o.I ILVI +11 « .TIT...«II§.T.I
V. .T buy IIIII1 ._11 11 I14 1 . . r I I I .. 3.0.1;
.1 VV 0 IV. ..IIt1h1I0 ll 4+ .IIOIeIIWIFQI? I 0 n _.E
I #0... 1.111.0«1011 II_L1 «11 1o ._. voqzwfx .II I ..-IIT.
I p -903. o I... . I . ._.. ..x
1 VIHMQHHHHIWW -I-. +11«.H. Eff”:
I IOIIIYIouI I.....I0IIIII r _TfTIII.
T. I. _0. . ._.?k. ..zlfi...0+vk H.99LIJLFAY IO1PIL1I.
V, . I. . .... .Iu I.IIQIIIIL .rIOIIIH4.+ITA w.HH9.HI
. 11] 1 IIIVI.-I11 ti HILIIN .. find- I... ..
I : .__I.....it.I.....Wfihnwut - .. Fir... .
Y ‘5; __ . ._ IIVf 41.. .i I “J‘L.+IQI. I I. _w o _ I I. I _V
M? F I Ali. I a ._.IO..I .I.” II" n .- WHIH ”Jul I... .UHIHIQ anHHIOIHIIII L+TL+II1I¢I ‘IOIYOlV “4 w w .— w w .I I.
I a II _II V . V ... . - . . . c. .
Y «lbw .H116+,9._.I0t_4.‘ .__)» 030+]. _I|_.I.II..OII.V. Told II1Y1HICO.XHI1LJT ....1ICIOIH0 OH .HH v "1“ or.“ fit II,“ . ._.
IIIV I..... _ . .. .. . . .wL AIMS: 411..., .I. .IHI «31-71) I I.IIIII-II_.T“ ._IL; . ....
. II. I t V .. I .. . .. .. _ I1
I]. III“. .IIaIIw I]. I‘.».LPIIIIII....: M IiiL..T. IflLYIOOIVrQII ...” __“II~ Ifi.
IIIVILT, .IIYleI I.I H I. . . . _. .- ILI+ «I! I_.QIIOIYIILT1¢ (I-I .H .4 .+. I . I I
I 0 CI 0 1 V v. I, . _ . . I .
IIVIHb—I II i. al.10- .I...VrL‘.lQI;|l!+.la,llhlILrlfld.In¢n I IfiLtIrI._LI ...-I .7..qu WI..- ##4}. I II_ ..
I II 1 1 1 1 1 .11 1 .111
1&04 I I. Iooa vIvOo ..., . VII I IWLINOIHflIIHrI I" HI 11lv+_fl IHIQHIOIGIO-llolotffllTquuTol a
I 1 o OLIVI.9II ...I, ..II . r rF ILIIFIFI. [.I IIOII. III....I.IIIIIII#.IL.. . I..
IIH 4...IYIAY+I ..11. III II.I On I.&... I.I.IIVII I” v. I_._ 40.0.“.IJOII01I ..VI.UIII’.I,'.III YrFI¥I' w .Ih .11 I .1 1 .449 I o .
4 .. .LTIIA. x1.r+9.—, ..11 VIII MT". .FHIFHI..- u .4.. 11 . ..I.IQ.--.1:r_+V.F IIHlf.MXW.HL ..I+V ..IohaviIA I .II_II_IIITIYM.L,_.I 1+ . 4..;
I ... . w or I In ., .I. .m.- IIWJIQI.II VI H9 MILVILLvtoy 9|" I rOrIkIe .III+:II.. . ’97,..44017 HI.».1:er|¢“IMLIII.f11IHI.I1Io.IH I . I. .... IA _. , nu?
:4» 41V 11 1 .I. V I I. L ..I.r JIII III. II‘. . IVITOO .IQ.OIOI .QIIT4IIO 41.. 1A I.I , .1 1.. ._.. ”
fiLYI _I I n . I. ¢ .I . . . II I I O hII ”I. I_II.L:II4.HHI ”0%.“; rTIlr . II .T.LL +II.II.L.I.OI..—IO.«»I4 Il'u..| A . I I I I I w. . L I H. _ n...
A: I. .... .—r. .. . a «I . .. .ILTVH . r F, fin Yreollr. .rIIQIIIIITTYFIITIJITIITJrf .. . 1 . I . . .. . .V .9 31...}. I .> LP
11 v+O IHIPHI. ._.. ._ . I ,I.1.I_.f I. «‘I 0] Fr hm H.IlJrI.. LL>10 w. Avrm 1V 1 .. 1 MIHI1
I > 0 I. Il_rl,I-I III If: ....11 HIn . OI — or. de.. LLIfiITfL..-%IVIYII_.VI'IOIQ ...» ,
«I I o .-I IIII.II_.-...+«.L_..TU 1.» I. . 1H. _ . fir a...» .. _ . I . . _ . a. .79.; 7..
VII .IIIII1IIII. I .P1IV.I . _._].1. .‘ II+II ~._¢.I I RI; 091?. I [IIIIIrIII
I‘I?_9OI.IIO:I I... I ,“nhai .I ... our. r.»ruot 1m+hvaerI I»? h I‘ItOIui .IIIIIIOKIHI
Vivi. IfllllIf‘II IIHVWtUiI..NIVH..IrI .I.—g Ia—I HI. oak... ._ ILLIT OIHIIQIl—VJHIIHJJ.
_ I. 1.1T»? l _ . --III»..:4. Lu; LII -I .IV.. nirVILIIV. _r; ... I..... . ...”.
v- .ovoI._n.I.+ I .1. _ . _ ... LILIIIHI .....o._II.Io I \Iflu1_ol.¢ _ .w r.IIIFIFItI
J a... . I I I I ._. LIF+4L +1. .II+1 .4 .9 V_ o. 61:... ...
IOOIO..IrIVII 7.. L .I.. _n [I.lIALPLIIII. a. _rI I. .¢ .w¢ : I I I.. r ..LIv. A: wLthw Of MIIH V. _.I..
II. 9.01.}. I!.|I1II|O.$ IIIIHIIII ..x _. ItmLVLhw .Ith I“ lfiIIII.‘LA x 1 04 {I I.
.1ny I 5.4 O... DIWW, I V. I l. i I . r Furltfiu . .. a III. ..II: . .: I . .4 . u1rI I .I. >5 .I,+“l Jyl: _ 4 4 . V.
1 . 1.11119 -..-.. 3.11.1. .. .."I . ..V_. .. . .. II -.IITIrIIL. .rrrw. IL; .49 .E .. ._. .. t :1} 4 4+ I
..A 4IamouIr¢ .. o.r . ....r» .. {OI-.I.VI...€I0.¢.1IL+IIII_LIFI_ cb.f b01003; OI..._r.F 1b .IIII.III_.
II .¢ |l¢lvl¢€dl LIIIO...+4: ....l H. .IIOILI HIAOHI... .IbI. I. I! .I.. II .I _I _IILIIY. .|..Ou...¢6‘1.1..+l4| I.IA I. . a... II . V
olo I.I..III.IO.I.4r a...I.I4I I- I..I 0 . . :r . "I _ III? II_lI I rrfio.’ F.._.o.”. ~ _.I.‘ arc.“ I... II IL H Ir....w.~l+l.,._,.
1' III-JIMIHIII_ I” IIIHIHI.uI.V.I.IVrHI.IO HUSH]IItoo.-I—II..III.1._ $IA§OI+ IOL..OI..I¢"y1 L HCIO .I F I.”
OIOI.,II _I III I.Ilu ..II I I 0.1 ..r.._— .III- .Ol ”IIOIIICOIO4I_.FI IIIII¢I¢I I..IO~.I..OIJ_Irr LI 1 10.44... . .
Y6 III. .I.-InIIII. .I.I .IIIOIVO V :L[Y~ly.il .OerOI ,IIOV., III... . YEVI ATuO'IIV..l.IIIO.IQII..IO1I % 111‘..I_I .I..)WI.
I OLII..J.|V_I.I VIII IFI—v. I..-....II.I...VI| II III . III ..I. , u ,v4IAVI&I .I.-I..,I DID (I.I..‘rfi .IHwI4 > > 0 ».IA I4.4 I.I-r]
IIITIIOIOII‘I'IOIJ Walt Fr b _fi H.’ } 4H» 0 0*1L1 1140111 __m
b k I'll: ..II. 111 111 4 1H .1. 1. . II. FF 1 14; vol .‘I .. I
: III. n. I. : ... ..I...._._.I.....N.J.._. ...w............i:r.,. . I r:
1;an IIIIHIIII I. .IHWHIWHIYH >09 v; I.. +__.Io .QIOOI_H AAvaiilerI+lf|ItTIQIFIhr.oh... LI ...11I. UJ..~I.I,_..,.I.|\..lu|Ir
1.11le TQ.I..IHF¢ I. VwIIOVIYI r I j.».l[A|'I0I.IOI.+&._I ”wog? I II” +1.. I..; ......III.O.IIOI6IJ+Ia $11.... 0..
r _1I 0I1I Iv 1 I1’.. -FIlmlrlii ._. I I4A A OI .m. I t; «OIVOaHM‘LrtVI _ _ PWOWWL TAfiépam.HI .I. d I ”I..IIF.+I..I..I1I _IWIII I —VV
TI ofiIIIJIIO‘v’II. ”.IIIIOOI I o 41... 01411 I. .Ief II Fraa I _lILI.I.V_.iII..m
W 7.15%..» . I. Io..$r*+..v I 1. 1.7L. JVH‘IOIIOIIIIIAA e.»II.rII'+ 114 I. .-. Trezroo. o I.
LLrIlrfIfi1I4l ‘IIIII_I1.II?II o bl bk» H LAVHUAIIITOI » lo.VtIhr.O ‘11} II“ 4 .. .1».1 I I . ...~
Lr 1 IIV‘IIII 00L“ a H1 H «r rfiofiv06¢ *AIIIIol 1 1 Ir...;. I. IH I h.
H H ..I ......TLIL V.5 5 HI1 H? H I. I. .. I pp». ..H112.
H HH 1 _. JI‘II: 1v 1 0 I» 41 1H 1 II 1 1 1 .34? .a I01..I¢.I.. .
b [II-Kr? ._ b .I II 41 r. .
H _ (I.I I 1r I ..II. '9. .. : OIfiA O 4 w 0 .. _vt‘ 01H 1 {I 4 I. r 4H . ‘ I I v ._ I _ O
... . .....f. To 1$.HVraoI. Ho: . . A ...:Vbr I ._.. I.» II _ I j..P1LoT..
I I.. »I I...-T 9.0.6.06 T+JI+I.¢W 1 9 40 IA oooefil1 1 evth II? “I01 P ‘IIIF. FOIWIII _ I o
11 .YFIIIOII ‘1 _ v. A c¥ I,[.[ I 9.7. OIro'fIIII41 1 IIIV II 111 1 I .
..IVI‘IJIOIOIOIIII‘QI Irl Y’AA 4 A I1 1|4 ...-L Ilhl. ,Llr ~00 I OI“.1I..O.'.OI I _ . AI
noocu.o.aoo. VIIOOIIIIIIII1II0L4k4 . 97+. 4 ..r64V904... 1 I....wéatw. I.IIIH.V._Y..111_4 H I O.» .Ooeovtvo Q _. I T
.1 1 1!... ..TTIII...:.;.. ...... . I I or 414.. H. .+r.1.._.ol.... I»: h. . .fr L191» p I.I..«oéo... . _ A U
;... .I.... 40 -I.. ofIJQJI, 0.9 V.A .ev Ii. 1&1 «...nf.r4.otlo1x.?¢L.II_§.: .9 H I+ .10.... a h
.. 11 1 ...... TIIIIIIIIIIIrw. .IVvo . ..AIII JHIIIIHIIlI coa.u*.....I.L+T 34:41» A . ”I.I.NJ H V791 . .. I
1.. .u. . ._- . I+f_-..._I_rI.114u .5 3-...44 .....Ir.-r_.h.:w II. .. . I . .
Mn . 1 1 1w1 m 1 «:1 Io‘ur+o$5hLLI+MIle¢ll1rWWAI I ‘q 1 Iwaovfia9 I
M o. H I+HA “ IHIIHIH1 I« tIvI we .I+_.._.Va_l..IdOIIOIOII\.IT . _ r”..II "II. n
10A 0 V 11 I . I . .QI I. .
o .9 OOAfQ . 0 ‘IoavPOAI HI 1 I1§mwfi “Inc""uIOJQ.OQOIA I _ I... ._.r
. .. .orI I M ... ....IATIJJIITQQIW .N . H _ IIIIIIII? .4.- .
I I .1 AI... IYYA'LIIITVIQIIQ pOII Irv ”O.»I¢ O .c
1 T v 0 1 I I I . In T ..‘wITIQIO I I .VYOI‘IJ I.. — vi... 1 A. I I I. O s .
...ILYHUIII. ...» . _.IL...I0LTIIL .. .I H+>v..+. _. I.......I.. .
11 4 on: I o. . . I: $1.4 . ._..IAI1. I-.. . _I 141V v.11... 1..., II o .__ . ._. 1...
VII I I o O I 0 4v 1'1 I” ._ I ViH I I. I VIC. II... III I‘ll. I I. I ..A I! [.I.v II
M I1 0 . 1.11 “ ...+.+t.17;. ...". I
I I.- __ .. I . I . III .IHI. .{JlIOI . .I. . . I . I .I_Io .... I _ .
.I ....I.I... T! m... JIwI..I.otlIvI filo II II. _ I..—.I..
. + I.II t _fJIOIOIIIIIfII‘rIII‘ v.valllf. III.tlrIQ.II IIIIIQI “Iv I I. I.
l on.tvoII I I..-,9FI VIIIIItIIII I I.IrIIOI+lIIFI .I ..II. __ .I.__r. V.
. I I .m I.IV r . T” 9 V. I I I .. _ I o I O YIIOI |.|I.l. ) .I..in .. ... m > I . _ . I I
I1. NJ I.I OIYT. II. II I....va .IIOI I 1.»v¥.IO.I.r..I
.IP 20.. 01...... I _ II I v V.IO .II ll .. .. . . ..IIOOI..I.. rt? 0 I .“I N I I
r_ _.I I II ....r __.I r r I. I . I 1.9 IIVIII . .. . .. n . , OII.I.. ..I_I I.I..I .. . , I I v I
.0- 9 D O ..I. O .I. .\.I...I I.I(l. t I _ I . n. t _' '- .__
III. ”..II IIIIIIILIIIIlffl_..'.DI¢I-II.ID..IIIII IIIIII'ICIII'III’r I I. I
. I I I III II I.IllIl I.I. I. l. .
I I I ..II I 0 I III I I I.. I .
I ...... ..‘r_ o I I. -V.I. IIIOIVIII . I VI _ Y... T . . toot
II.I IIII . +IIIIOIOIO I II . ‘jV V ...HIVH
II . VI .. l IOIOIAII ._ III I I I.OOIIO..I..4I{_ V "I +
III I I . a ... o I . IIIIVIIII 740911.... e H Iff
.I. IHI.Ir..| . I I I I I ...-III 10.4.I—IA a. 76.
I I I.”|I‘0.O II V. I.. I0... « IIIHIH. aI~ ....I. P
. . H . . N "I .rJ. I OII1 111 1. 1
.9I I... .. V .. ...I, I?! I. . 1’. I“ .I.“.I.I. .. . O. ..II- II”. D .Ildmlo II‘aou r “Hal...“tl
I . ..I 1,... . I ,0... . .- IVIO IIIIIIH 1...... . . .. I-.-.LII.I.I.I.l. _ . . To? ... .4. 1 «11.....L:
I. , ._.III I If , . o .II..I....IV. ....IVV_I.I.I.. I. . I.I ... I «I.I. [rloawvvruwoengIILII
I III . : .I . I .I “.I.. 1 I.. I I I vaw4IIwmIJ. V+H~ .rII+¢
10 o o IIILAI I I _. on». I ..I I. ...-4A.ev. WIvovlr»
TAVI I III. I I I I I I . onlbctov .Ho. . kIOOOIvIvILI
I I ..IIII In I. out. o .MpaA V}.
II. I o ,I I 1- I. . .. u. _rhi. IIHrOIWJQILIIo. 5T
III ..IIII4III IAIIII .I.. I.” II OrO+VOIVIII Vol.1. ..II. A;
I PIIOI 8... I I I V,. . L. .. . . r L .
I I?! YI I.I I u...- .uIr.. .I L >0 HQW"%I JGFOMOIOIOJ_ I++M+t+VH . .14II
. .. .I . . .-- _ .II .. I19-.. .. . I ? I»
111v?!L HFIH.II fi?.-... I. «wrfi+m. .011. pL I.HIII'IIL091.I.WI .I.. rfi.rHH>.ILLvrp0
Ilrv.r.¢. .II .V . II. ._.I. I. ! ... IA HLHIIIEI . o.~o H .90. rworwIIrIIu§§1
111.17... I.. I IIIV..II r. In... I.. _H IIJI......F.111I.11.1V QwHIIiIrI
ILYAY . I.. I. .I.» a IowIIT+é 11414.. 1.111.?
IIIr. e+. .ovvicao.» +IWO+ I11Mo¢ow “VIII III
1111' to... VII. Oafi; I+ Ia. _V._4 ._.c_ I _IfoI 111
T-" .............L. .... TC...» .Ium.whfir.fiw.VVVw.
... .. ... ._.-I ....3...VH ohuwafrw I1 III»
111 I II .I . _ r
EIOIVI. I... Ihmi “Iv.- _H.I6I.Il?.o. A¢\_.I.9.HIIV
_._...II III... . 4 .... I......H.4+¢LT4A10 1
.I.. .IQIIv‘Ico I... ..II. J? 75+
06.. ..II. I or... I..... .oooowf . JIWAévll 1
II .er.o|‘. |<I¢1 III... IIqIIOb11I11I4
60.. -I..III1...r . +IIIYII- 16?. I...”
... -Ir... I "I1. . .I.
..II! _.tw-.£ ...:Lu. 3. J
o. .u r
I

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

DEPARTM ENT 0" MATHEMATIC.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

     

 

 

     

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

     

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

      

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

UEPA RTM ENT OF MATHEMATIC.

 

 

  

 

 

             

    

 

 

 

 

     

 

   

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 
 

 

 

 

 

 

 

 

 

 

 

 

 

     

 

 

 

 

 

 

 

 

 

 

MICHIGAN STATE COLLEGE

 

 

 

 

    

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

    

 

 

  
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1 44 4
1Y5. H H000H0H <41 44 4 4.14 4|4
#4 o .- 00.00000 ..0100.00L400NH0»H77 «00004HH 4d 44 4
nH NH a "4 0 .0.--0-.- 0000.011 V000 > 4. oJH¥0 4 H» V HHHHrr HH 4144 4 1444
44 400<.-. 00.00 .b.|00 ... co ..4 I 0.00.0..F..0 H H 4 044 o 00H 144.” 14 A 4 44
.. ... ......0 ... . ..u. . ....I....: .I. .0. . ....0 . 4 . Hm » 4 . 4 o. . .0 . Hm I 44. 44 4 4 44
1.3.1110. 00.1.0.0... ....401.--oo-4.00.00 004+ fain0wo-0Hu NH.». 0 v0.0 . ”W H 0-.. <4 I..]. 0 0 .. p 7004- 41 0.00 ...-0H. 0 .4 144.4 4 44414 I444
p.7010o...-o...to_wo-ooJ-.o...o.!0.oauo .o+.«.o» 4.0..001.I.IY1..YH.H 1.44 4 4‘ 4HLII4 W4WIMIH. n..4-........0 o 0 ...A .0 mm. 40.5.14.-- . . .40 0 -. .4 .00-
... .0.0.0000 -Toyvr. _c‘_ w .-"0_ v.vo_0_044 044 . VII... H ..PTI‘IIO30‘011'IhL0 000000000— 0.0.4....” 0-44 I.. { VY VII aIL.-I’..v._u.0.h+0li _ . . .
... ... 26.-.114.“ . o...~c0 I 0.0.. *4 ....+.... . .._.. 4 -I+40...00.«4oo... ..:-.... __.fi...nu.._-. 40.04.. NH... -9 .10 . .0..v--01.-;.I0+..0r .MK 0. .. .._. ..
r . .. _Hfifinlolo; ”.944Ww10-70I01LI r00+uvr¢er . 01v v.74... _- .....0- W005}??? my. 0H +4. ..:... ...0.--.AY0.N.0-0..0-. 0.0- .0.00... o — ... 00.—..«0 0.4.0.1....iuJ. .
a 0H ._.-0 _ -7.-4.fl..r . + 4.M’04H.VTI OIOI1.W." .. .0 0000004. > . F >H9+—. _.4PTO.» 9.... 1.4...+I‘_OAYOLIF‘ 0' 00_~> m» QOWIH or ..JOIT. 00..
0 _. ...5. . .. -, 4 r > .. . o o w . . . .. .0.. r. . H . .o .... . . -o...-.0..0._-0..
.— 1..6f.9.6. .... . . 144 “—4 I >7». 0 .. 040,091.04 ...w0lt. «L 00 1 Lrl. 00-0 -_ _ _
0.44. .I.. __ 4.., ...._.+..-!0.o. .0....0 .0 O H. P 4 t L- _ _ I.I... A ... Ir! ... .1 . r . .. a 0000‘ .I.-9+ 0o
0%.. ”Ha 0....-r_a ...-“.I..“f -TAAVOiL0QOWHH-1 1 .I.-.IOIII :00. . I¢0014ufl ”j. 4... H H H H; "v r%0 ~ rc.«.fi..r ”0% I ..+.o.0L.l0!_.0 _OL.
4T0. p.4 _.0-0' ac... .4, 4 7H».-. H.» ...... I...70?v-.o. . .. .1 .9910... in. .4144 .9 N m o H“. 17w...“ . 4 «irrv _fiuJon ...-04.014 I .. ,1
o a; aW¢0o0I04o..0-0.+.-.0-¢.9. .+.¢..0.o¢ 00am 1% .4.-#1700000. co. 0 ...-0.0.9.0“. “Ho-FHHr r 4. .0. 4 .g o .I.-0.x.stLHHN o ”4...-HI... JEWItbArflq '0
.0 . .Hflinflot- +._.+.o.$l_flrI§.-+00ao..oo .Ifi-~fiv..?lH-. J v- 0.0. 00.. . 0.40000..w40.14.FW. ..rH" .4 4 Ibvna .._-H 0...»? EHkovvo V”. I .0 l ".I .00.”-.-_-r+|.10.0-.0Ir
.. . _4MH4L0OLv0. a". ._...0 ro++ ...-.0. 4- .1 ..Lo4... ....oku-.¢4 4- . L4-0r 4. I Y.-T._00.-.0.w+ 0. 0.7... ... . v .. . Io... -. {.4.- ...4.
.. 4.. . 4. ._.wovO .0 00.00.}. = IHI. 4 0'th9 .4030 u. 0 _ Iaif‘ ..II IL .IOr -...-.JYAHI. *"H _ .I.v '0‘0 . .111" .0. .9000
. no ...-H0v0 ._....0 0T§0IIYO0. +ovownm H446 075-0.- VIP} .0_14+r6.rw»-. . 0 .7 04+ 4. ._‘oy0 . _..r_14000...,- T00. p.-.I
. . .D -0 III . . .... .. |
......r..H.-.-+...w-.- .-.-r 3.7:.-. .... ...-..: H411... . ... lift“? 4...-“.111..- ._......+........ I...
Irv} P . r! I _ _ 0 . _. , .. .0.I.§ I I .. . .I.... _
..:-1.2 . i 4 .- H - .. .... - -.. ._. ._.... ....H .4.-1...... .. ..
..4. . w.‘co«-4 0. 0r» . . 0 o. e .00 .. 0. ..- _.
.. ’04.4.o.0.0.0.7§ +@0.oooo. In. r H ..H 0 «Wflofi . ._ F H.010l-0'0.|0.0- .0400...
. ..-... ...-r....-........ .. 2 ..:...“ . . I. . a... ...-1......) .
..+.0...Im...+_o._.__.: ”.... .Lléng. ...... Z r... . Turbo...“ ...... .4.“.
0.‘-0.o04. . 0 .. --790.|D0 0000” «a 0.10Imli-r009.01 I -0 “0»00_aw
._....HO .4....- v|0 ITv’fi'ObOv 00 .f .IH-ovo .fi QQHOQ'T. .1 .o ‘ >.v_0~ _IOJI
_ .0.-..I_.._0ou .0o...00lw H v.1 __.00 00 I ...wo_..o_- .._.- -,
.... ..04’-7_.V .' 0 14 ‘0 L . )0... no... u 4 I.“
._.--..: ... .. it... _ : ._--...- . .... . ._
. ...; ..T. . _ o -- .
11...... 9-1.1.1... iH .. .4.... I I _."Hwn. .. H H_.__.. . .... _
IIvYIAYQ . Ho...0-.....H.0*»..90.HI0H..> 4 4M 0 0.“...o_ . ...0.
II 0 ... 0H3 o ... 0.... . 0,4 III-.1 a. . 0”. ._k. .r .H - I
. d I 0 J .4.-0r . i .1 . .
IIIv0fi rt.4_+--|--wbg0*+.00¢9W04Nw 1.1 .ovw4v01vl..v.t .1_‘.ti-.IILIW.I r- -_.
{111101 ‘$~.oo_on..fi._u o-v00..0I00H4 T000? oaqu.0 » . aw..c.ro.oo _» I.
0.. .0- . o 0...- O ..I, 0% I.. 90» o 0 A V 9 0 9:. » _Ofi o o . 0 I . fl . .._
H r 4 to. _ . 0 .. - 0 «4.9.4 9 u 4 H _ o...00.0~:0 .... I. 4.
H0 .4 ._._.__ ‘0 ..+00¢.. 19.99-0000'} woo. - .7 4 0.0.0.. >. _ .
0 a. 00..-,4IT .. +u0 00uwoofiu0 40.0.00 7.0.0 .0 > .H .0.?- qar. ...
*0... - v.40... . ._4.Uo ”.00000'p 00.0... 0. 0 _ c..0c.0...>c.. _.
141$ ...-l. _o. F-w. .*.000.-0 *NI . If. 4 000.._“%0001..104I0 p.-
fl'OfIIfi I..-l. . __- ”.1 b H .Fww H o 0 00¢ o.m‘w+v.0- 0- .-
4~0L~u~ .. ._."..-r.. pow, . . 4H4» 0 « I 00«.W4._...i._- 91-1.”-.. ... l
0 l0? 1 ..:-I.. «:0- u ‘, 1+0... on ”H0001 0 cf 00-.ch 4.”00_. L.-.A -II.- 4|?»
40.0. .+- 4....000-0. --.V.-0'9.000.0.H.+00ooo.no. v .1..HI-O-t0.ov. 4r» .4.«V_-|4[40l00 0' r.-
+_0a 'r . .0 .... 0.I-l00000..0..0. _. _ ....0.« _ 4_«po.l.IJ.¢-llr I.
a a- n. 0‘04... . . II v. 1. .
010 Our ... 0 _ o0-fiul. .-OIO-.01.0.00 0. 00 6 0“ wiaow‘O. III .0. I? w' .0 0..0 0. _
9...“... .0.--... . . 44.0.0. I _ ._ .F . .4....T901ri .. fvr..+l...-o f0...»- _. .._.
-.Llf. - . .... ..:-.44... ..LJh-zduuw ...... .._. .114 0.«-...¢..--..o...... .H”
111v OIHFO ..0. .I. . _ a-‘ I lLJ. _FIo-t 5.0 0.0 0 0 _ .. t o _ _ 7:.fllv.’ [0.0-0. 4 . . Lw I ‘ 0 ._ 1.0.... ~70 + I f. A a ._ ..
. ...-.0. ..r-....-._.. -04.... .....fi 3.3.1.2...11440ITH1 .n... ._ ... , . . A
III ._.-" Pub]. II II nit-..II..l.--- 1......- - _ u . r 4 " .I....J 4 _ V- |.|0 _r 0 0 0 . l...
IIlHfioo. ....” _04.. L-._,.-I0.14Io.oo.o-u+-HIH-J...M .. 4h4... 4 H *4 4 o m ... om 00 ... _
IlllvkflLIH<Itl0lfi 1.0."..an - ..Iouquo ouhatwau0+ “F. o. vor #YV4. r44 m~ v 000 I..-..: 0
1111:. 40.1.... H, , .4.-.....04. 0.0 0.009 r... H“... w .I. 7. -._r. _wL_ I H. » .... ”I 0 .4.-r .00.. .I0J...-l
III. ...rar .. k1 .. ..._-.0. 0-4.0.510 .._-.Twr0.T...-+ *fi.... co .7... Iv _~ ... -. ,0 . .
4 .I 1 1““ Y... ol _.J. TMIO ,a 0 0 0 f b O 0. 90 _ fifi . 0 o _ “I“ W... VI 0 O 0 no w; _u w c 0 M . I .d O I 4 - ...w -L40I
IIIvN¢.o_oo4 .._ __o.lw40u,-.Hn-00.0lo.00.0.01..+. w ".0."..001 ._l1041 I0J ‘1}.4 .... . ..
b .+ to-.. ..-_-. ....o" . c c .__ H r«_Fo .000 v.H.H 70.9.0.9 V .0-0-a0. ...-v . -H
. ...” .._...ITrIFTrI L- L ._n .. 4-... s - ..- w .. .... . .. .
r. . .-...- ..--..-.- ..-.-...--H.+J.t...t. 1...:- T N _ . .- . .... . ..
_41 4+ ti)....I-.Illt_ v.0 . .. .I J u ’w .. C I . at W ‘ .I < . . . . 0.4.10010..'F-‘.IO - .4. 0 +.’I0 . n 1 N . 0.1 I It A .10 I-
_ _ _. o 0 W10 9 0.0.0II|+I§I . I l ._ . . _ _. 0 >D 0 r 0 . ”M . ; . ...-0.0.0 [HI;| . r. .9. .0 o A I 0.0. 1 _
..:...i.-.3....-.»--.....H..1..3(Hilda... I..;fl . / J»? -..0.-............-.... ...-:1.-. . .. .-.... _
.00.»;9.-..H4,..1-4..-“”H_. ” .. .0nwshc.v~.ogvfifioo {“44“ 4.04 I H 4 4.-.0hq.fi_~1‘00 .4.->10...l._o-..0lnll_-.- lo 0.0.0.5.: I.I-Ill! .0“ . .._-
... _ .o .o o... -0 . 44 4 >4 4 . 4 3400-000 I. _ .- . u. .-.--.. o o . - -II
..-..." .++.!......‘ 1.31.4... ..Hu. ".I....Tlf..f... .. .134qu . .- ...-.. L- “Lt.-.” . .-m. .T _.
a .. _0 o 0 0000400 I «- .0 4.0.0000 0 0.0. 00 704441- } «0. .1L T.-.v0 00?.705 0.00.00» 0.» 00 o O . .-.-.0.. .& .0.-I 1-.-..- .-
..t. H. o o c 00 o OO‘LVQ..IO 4.vl'.H.l. _ Dfilbl I. . JV 14 u. I ...-“Li- 0.0!.» fl] 04 OOH Ito-1|. v n o 0 ...0l..\- . I 1 Y.
+-.-Ho.....1 :7:: ......»CH4 ..., .....000-.r..o-s..4.?. . no- I. .71.... ”...--..”...Q. . .. .0 a j
. I .... ., .-:.... .0.. 0H4+fi ... ... 4.0004 _ run-Tl. .-_...o.u0.......n .....0...—10-.r4-0.?0. , r" 1
JH" VIII-I.....r".IO%. . .— _ 0 4 H 0.40 n .0. _ ..'.U00.0.-_fra......0l.I-, ..I'G'..|.....0| _ _ 00.. r0_ov¢O 9
.- .I.-6| ..f.-4..0. I..-.._. .. 10.0.IO ..L‘III“ .% “o vfi._ } 44 ...0....FIIOO.lnlrlw...ll,'..+l.'.f .. _Ifilfr .fl-H I...H.J|O..II O+|0 I60"
.9 .0000 0r-0Io 5.015.?- 00L. y a? ”_ocV0. 0- BLJII-.- 70...-.0Qillblt 010.00.-.. ...l. ..v ...-0. 00.0. Y.
.40 000 000» .4 .o .0“ .-"rU-_" H“. .0004». 0M4 . .. 4.-.-0-.-...0rkM.-OI..IH0LO-0:J:NY:k0....-00+0 ...00owr.r
U .0.. .0 .I- - _ . . fir .| . : “I l0..l.\. .I'la .... _
- 01000 .0.-0|? - - . . .. 4 . I . .__ . ....+-0.0.'J¢
art... ..-.-. -._.H40n. ”unnuafi.” “ H . 4.4-- ..:-0J4... .Jouuuw.wwu.m0.!..w.b&_ u :11:
., . .. _. _ VI. .. ..II-0..
. _I+.o ”....- -.¢ 01.0-90.0..- -0|o0-,.o.oo m. 00Y409HI..II.L0.HV - «a.hq-_.1+._v+.-0.0.dn.. .I.-0 -
.30.-. . 6,.-1w,+IIH0..”H.n ..:”...oi 0 Y00.4.06.TH....HH.4T .....«eatv W...F .4.. ,.
II.....W1...... .-:....-..14. .H..H. ..N”.v¢ 0M000 .1... _404 “Iouk «+0. 3.1.4.410 I.
{7.0.0140}... 0101.10.11.01 .I.. it. I'I....II|ICI.I.O 1...“. n . w..‘..l.:.. 00. ”...-OIYHIHHFAu-I. . .IT 0. 00 .. .... .0 .0..
410.00.140\.0-9i I -.rwm...” - . . z... a,.FL;..MLH0.w000..M0-4»u40_ ... 00 0,. .0 M "an“.
0 to.» - g . 4 .. . . .. _
000 DIMOHUw. o. n_-.o_o_04. H H 000100 «.0 ..._~+¢~HH0 . _-IIH”...0 .. ...«g..0..00.‘
- .. .. ... .... r 4...- .I.- ...-4 . fl. .0.... .21.
u .. o ...l 0.40.... 7H++ . H 4144. I -.....
. . _ 0 00.0.0 . 1 Y? ...... -I.!
- 0 I ADJ-r. 0.0.1.00f9.v .. _ .0 ow.... I.
'00 0. u v ...-...... _. 001A 000+000r. .000_ a _ .
0.0.00.0 0_ 00- I Inc- 0. .-.$4otr - n .0000". 0 _._...r.
.._0 ._0, ._ u 0.0 .0 .0 .u.1u.nnm...0 .n v '00”...- u —m-r,
... l.’ a 04-0 .._ 9" , . .. . ... . .0. .0... o I
. ....I u, _ .0 .0.. . PI , . .. .. . . . . H _ 000....aw0.‘.00.;0 . .100000u. — | . .
_ _ _ I - 4 .0 _. a _. . . - . .. . . .I0. , _ __ d o. .-.-0.0
.1000 00.00.0I 0 .0 0 ._ . .._ ....4 U... . . . ... - . .- _QWJ.0-.I._Ltv0.oA:0 III.- .I'nn._ ..0|.. .0»
.. . . .. u . . . . . .... .. . .. . .. .. - I-I.ul _ I . . ...- I \L .
..., .H .- _ .. ...... . .. _ . . . .. ..--...-.......u..fl.--J:. . . . ,,- -. --- ---...
Iv0000000 o. 0 o I .. .,_00-OAJ.VID .0 I.. fi-I 0...."50'110 .. o. .... _ O .i.. , .(ulw'.l_lln‘-ILA-.
VIA . o .. 0 0 W H_ .0 .fl.~.0..lfl.9. .Oli-Q. IOIV...-' .... 00 .0 u .0 [1.1.4.1.- ...Oo..,l”..... IV.vL._ _ 0. _ ._.oLV. .0.
. 4 . . 00 0. .l - ..- _ . u - _ . . _. . I. I .rlr ,-
Illoauoo tn... 0.. o. ...w u .. 000+ ...- .0 0.4 . .... _ . .._ ._ . .-0-.4pu....0.0--.. .-
I........... . .... .. ... ... ... .. . ._. . .. . ... . .... .
> _ -J . ..., . .I¢.0-O _r~.. (i... L 0000 1 . 1 ._0. Q... . .
L1 >444 I.I-XII I . 00-0..9I1m. .04 n. _ _I ... 0 _ o . n .. r o ....“c ~..-1 .
.0.. u 4 J. . . . 0.. [0.01.1.0 .I 4 . .- ...... . . _ o _ , . . __. 4 -.
.170 0.- -0 a.” - -. .0 0.0” 0. -. -.u 0-0.- . - 41.4141] -I.. PIHI..-” ...-$0 .00 _.¢+. tiff... I . I._- 0- _ __0. .0I0IF00 ”- _ . .m .u... ... a 0” . .f.-. .I . . _ {Iv 0;
$.I0u a .0 .- 0. 0 I '1 I. . 0-0L. JI'IOO..OI¢.Q.. I....HIllolaI... - . . .I. “.0 - A I. ..v, . _0 h 0 fl... 0 0_00_000_ - .0 lvil 4.0.10.4. ...t ....
IvJ-O.0000_010.I . _u . III. I a .0 0 . _oYvoIIOI.|.IuOI 40 I. u . ._ __ 0.. 0. . _ we “0.. 0 I..... 0.. I. 04. F
..:-4.00 .... . _.0 .0 0 .000 o . ... .. ... ._.. .-.-YHI'I ..- ...fi 4 ..4444 4. 44 I._ .0 '0 ..I .0- 1.... II -_ ..lr .... .a 4. *..:.. .“9
v0:- 0 II I . c I J _ o 0 _ u. 0. I .0. 0 .H .N H 00 o o '0 0.4000501... 00 000 o u _ o u I wI. ._.. . . .I. _ .I.-I.Iv-._...r.l¢ ilwr__ .r H . 0
I I 0 . 0.0 0 , 0 ._J 0 0. . .0 . O 00 ..0 0-0 . .. o . I.. I. o o 0 v — . u n . _c a _ o o o n o ._ _ _ I r.» _. LIP-.0.}... JIJI1-_.-_1rfiIfi.H-O.tr0.tvlf1
Y .4... . . o .... a-.. .0 ....0-0,..0o 0 0.0-. o u . _ _ _.4 .4. . .._ 0 .0.... 10.0.0 .....0..+...0.44>0
Y... A... n _ 0. . o ._.H 00. o 0»...Iun.0-00. .- .0, 00‘.- o . . u _ .._ (I..0--I..ra|0--0I.0.rn.0000..111
I - .0 _ _ _ . .0.. - 0] o . ...OOI-ro - I.. I .0 0... _a0 a b _._ . + « 000—*f4fdcl+uI-TOIOI r
.4444 0 O I _ . _. o .a-..100 I. .4.. _.u. . 00 . u o o 0. ... “ ... . ...-0. v $.14
v. u... up a O I-I..“1I1J-Illva‘ll - M I . .I. ..II I I00. . 0 _ .0“ . .o :0. .o ; II. I u a 0000.07'3‘0’b
v.v+.+.-. 0-.0v_0-.000- “ . _ 0 ._o_. u... o... . .. ullewtt 04-900 -.ovI .I. 0...! f -0 0 .9v.o00.-.. .0 0. . o » o ou-oono-IvQ.IIO__ :71
fi.ro..+_. .. ...... . ... . .. .. THU... ._."... fi. . ._ ..-. .. ._.... . .1. .. .... .. .....Itrfzo .HH. ..H: ..4........:H
4..-100.0 000. _I I . n r... ”H00. 0000....- 0 00.».*. .19. 00100. 0‘ ‘0‘ uLIIIv I'I... 0I0o4rfl {-fi .t-o_-v. 0.0le 0_- o ._0,0-H0H_0-a.u 00000-0hu-h 4o”_cr Q..
t .._..r.... .0.... o ... _ . ... . .. ..-. .. . .. .. . ....I ....o...-.. .. a. . ... . o .... .0fl!.....0.fr.0.0.0-0.o.r.-.0.. 00.0. I. 0 _._0m .0... .4.. ++rk¢LITI
VAcvv+ . u. a co. “ 0..0-_ c..1004... +-+0-. .0.. 0040 o0o_ O .. 000-o«!00| vo’aoIvI..00 _ _fio__t..00¢o .0. ....+,. 0014... Qll- 'I.4. 0.0M 0 Av“. I
r5.w,0._00 ”.0.. _._ u ..m c... who. 0 II.I 0.010_.+40000 00TI0000_0-00-0 0o 0 . .....1....T40..-... ...;u .r. 4......4. .401 4..? - wk”!
Iv 1H- - -0. .0o_n--H_" w ._..0 ... - 04.0.0._..TM-.I;010uo-o .0. o o 0a...0o...4 0LT0.00..-1.O00-. .0 I ... .. ...H _ «00. . a— ._.. v.» . .0.-“o... Mu+7 r..0000+.I.T.Y.l
Y0. v v 00“. .1 ..1M.....0.-.I .I.; 0.. I 0 ._ ._ .. h) _IOI I-OI0 O .I- ..u-I o o- o. 00 r 0 ”iv .I. .0, 40% 0 0 0...... 00. .IW-wfilo 0.10.0...H. ,Od’wlb..0 00 - ”*.IH"H fi_ a r o o o o 0. c
p L a». LL. ..0.~ I .... 4 L-._-?0 F r ; 0:1. +Y:-0.00 c 4.. . a... c .0 . _ _ _o 06.00-01 ...-...:- .00 _ _ 4 o 000 .0... L.
< H H . 4 1 4 4 m. . ., k .pl‘-.. 10.0... if! 4..... I'LL-0|... ‘. _V 0 _ a. n I. 00 0 1.: .0- _ . 0 . Ola 000 a +v 0 0 00 r
44 141.13.? ...bflwP-Pt C a .. . .. . ._ L. 0 n o .c .0 . ... . . 00090000 0|...- rI.00fi . 0.H+ I.
v...- Omfi. «a. r_.-. o_.__4_.~.4 P ..4. MLIITI Mtfiufi.‘ MtA0H4LF h>hr> Y0;V,0 .0000. cc o-‘be 000—“.muqccc. ..IoooIoI oo+0~.44..vo. "UNI-O: 09000.. ......h
v... .vo..v.-0ru.+_..ar -o_‘r..en.f.a._w4.o .00....v ...... . a 14V 0.0. .0-. .u.... p-.0c-H I -. -..-_.oc. ....... fi....... 0. 0.0m0H 0.
I I4... .0.. .rrvnlr 000 0. "LII..on .. 0.0.wa w . 0..E+ . o. 4.. If H . rI-u H170...v-04-.T¢.>.IQA . no 0 coo “to.l—_v..ll
IV V .01 .0.. ,. .. 00-.v090-0I0} 4.? 00.0 I. , - H0 co 0.... 0. . ..Id- 71....” 4. 4 P _ FPIF00or n I.. o o u . v .
I.. .v octfi a. _mflu +r 0. octao; J Fofi. 4.00.0.0IH.M_0-..H|9W.Wfl0._ 40. 7.0.0 ..Wlfl..n 0 - 0 ...-.1030.“ .0 J..L...0.4.40.;..u. H44 4 4 ... 04. A
7L0. -. . . .. 000001....000-0 00. _._ . . 4 D 00- 0.0.0.0 . 0 . _u _I 4 «0 0.0 0..r.000_ _ 00..-. .0 0 o..«‘..
71.04 0.....1... L _ ... 0000000 .v DWH w _ i _ 0 r. ._ I or 0.00.0 _ LI
0.. 4 ..4H...-.... ...-.....45..+4:-I..:-:17... «I..-.H...-.-..I$II...” ...-. .._ ... ._ . HI“. 3 . ..H...-II...1+. . ...
P .I- 9 _I.I... , _ _. . . - 4. 0000 n o. _ .0 .0 _ . . 4.. 0 .0 __ I I I.. 0*.--040- 0
HH...- 0 wW-oo 40.0w” H_..m.__ou ”MUN” _WH+H00- Y rfiowa00.. _”+oH"HHH“- -% _ 1.0-Hr. .0.-0r00. 0.. u. .._rvfunur. .I.—.0... woo .00.”.0 .Hfir
44 r» .. >0 . 04 _. . . ..I0._ - . . . .. 4. _ . A
«T; I 4...... .u Q . H .04“ m ..:»...I; .I.-f...» - .3 v I..-....-Hu-oo.+_;._.._ _._._ _ ..:...” ... fl: ._.._ ~. .4
o. br—O _w- ._ .1 ~ .34.0.?0. 0.1+009_H_-.W|001+I.Y44.H_._ I N4 H a. 4m ~I. .._..I._o .90 o “0.00-0Wllvoao-90 -0 000%4.. . ...”Huaflfi “”0." 0:.v _4
.. L T. p w -_ .. o. ... r.- 0 a .....a a ..oo 0 _> ._ .«. .. o... ..:I 6," +90 . no 4.. T
..:: I .47 v . ..IO' . . o 4 00-. >vol9oo u . . 00 . . .... *_... _. 1 ...‘I4-101£-Q.v-00§'o.+ ... M
.. I -4040-0 ......¢ _ . . 7.0 ._010 CI I.. 0% -. .-W. _4 ..e.... _o.-._ _ 4 4 . .I
...i: 3.....-......r..+.iv. ...”:L z __n. ._....H... ...-.....414424 4...-4....-...-:.... . ....n..-.."-_..:... .-.
. . _ 4 1.. I 0 _ . _ . . .¢ .. 4 4 .4 4 4 V
0.. v».o__w 0...I+.4.00o.+-0l'.vA.O.0+00.0.0.vJ0. . .0+.0-4 “-0 900.0 .._-0.. ..n.fi ~.u.-_ .0 . _.N+ 00?..0-kfi0trv.» P
4 *. . 00 0.6.... 0 I.I-L“- & _ r 0*......00... _. Q o a o .0... .1 0.. . 1. 4 4
.... ...: .0 « ._J-9.-...- .-.. . ..4._ .. _. 7.16....21 ...... . ... .3 I. .
Vt.- 0I .0..-r._,.0 o .00dfiac oa0odooo. .0 .briuuvué ...0.--.¢a_.o.0.+0.¢ ..00 .pL .9. 40604-0. 00 . . _. Vol-LN
hr .._.wL... . .HP—.«._ 0.50- .000. 0. 1.49.”... a FHL r.u-00.-.0I0I10.I¢.,00 0.....04.-9.0.¢4 “-«4010Ianrnf-.0-v 01.00.11 .4fi0
0H _ r-LLn. ...-.0 .4... 4.0¢ 04.1-00.1”..0-f4.._-_.0~0.o 1-"? _00000-Av‘t.00.0 o N.¢.H0.0N 0...-IHI.HO.H0.40 T0.0I.. fl. .
. It»-.. . 14.. A..- ...._._. . - .- .4.-.4- .. 4 ...o- .. 444 -0 0 ..
”-Hi0teflMI .0. _0...QI_I‘..[ H+\U.Q.L ‘411WNW’0 &¢rm“wI““ .0 00-HII’Q 7.00le Y A.
P _ .. . . __ILv . . v 4 .
. 7 . 0 Fr ”mm-H... Minn-._....Ia .0.-0.0H...0 H va-JQM-1.1HI0«
0L0 -p r.» anh HHHH

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

     

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

MM 00-“ o . a. 0.9....0I'INOOOO O o Mfiwl no. «1 “04*. .I.‘ II,4 4.414 n “1‘ ..Iv.0 #;
O. oyvnoflf a .0I0. « ._4 o LITHLVQWAIH‘ . a. . 09 1.00 9D F,I I
..L o .9. “an. ......A .IYY WTH+L Ir» b»... ... .. . .u.4.. .. o ..A
v7. I.-;I.o.--,.1. ._. .uf‘ 4 o.+.O.-o 0. ...Ha I'IQJIonI . .4 .- .....oo-vOA
QAT.ro..v*. o..... , 0 .0p.. v .... ... 9 4. r ..-... . -I.-v\-.+‘tl4¢17\,lo.vt
. .._o«,.o_r+I ... o .«tav ’41an .1 4.4+»? «If... v.0. oco1¢.u...o..‘
.0. 1.0.9.1004. . ... o IH¢o¢Q£ILIvJIovIYa Lo. o c..% .. I.IolnnA ..II 1.0.... 14.... ill;
9 #:TvOO: .. O'Ol .O 9111 0F‘10 r IV >4 [.67 .._.ab.’ ...Icl can... ct. 4 Q 0
f+o ”...;AVWLI- .._T. r on». o H. I VoOOQ» _ Lv MIL .IL . . IrPOIQQIIAIWI.IOIHN.IIH.-oIfl F A.
v u 4k— 44.41:” . MN... “on. o L p.» 4 > Iri .‘t1lll1... ..II-01.14 tr 4 >
4 vc-oHN.-.Hoto.+.t AFC»..th w__“5¢nofiOLrAwo+.oc.. . on. H at. . ..‘Igii. -t.l ..0| 4... .ilAlfiud.I.lHluMWI4r
r... . .r‘.,...4.. I.-..Ib.+L. ‘4‘._L.....vafvlto AA. : o. v¥¢owT w.¢. _. -Io...;.._ov;..~I 3+
I o +_o+iL|.o4IIII_¢.AI--IJI1.. O..-L...I+0H.,A. ._H.. a. o A 400L401...’ 041 .._ _, ,1 .II..+I.I.IVI IWL*u1'LnI.'u'IO. OJ
0. fiLIAAJo...»-..Mn..r+{#:hon. 4h-.. -.-ha+a¥””9 on «u . _.-.>*_.. f ».. , I.. H§f+._v,v.lw .00 19%
0 AhOL .‘wo .- I.. f val.0., IO¢I*I41.O.‘LrO Q «..Lvfi t . . v .. ,. JI . |+. , J . ,.
ma‘. A v“. u _L ,._.nw6*.-.OJIJ.44A_-0.O,O¢4.0.+ ..«v ....v. _. rv. o .. , I+,-L.Ao-.r .
.._“. a, + “I. I.I}.Oé 404.4 4.. . 4. on m. _. o c :49. .4 MY o . . .. I .I.... 6046.
v I.....IA.¢I.AI¢A.1IYA_Q ”gown... L; gab” ooOL I»-.. LI... 4 . 4.... .0.. hr...
1. IWILd 44 O... .t..» 4M¢.._F .._vp§lne%.o. w ..o‘. .vba....
» r» .>.. n r Wt pl» r 4 . .I.-[III In..._'ll rI‘IIL
v ”thof‘ l—HFIH...H.IH.¥IOOHMH$O9 H+ . w... ¢ .. <v .... ....L v.19... .I..-.
o o o .0 a .0 FOqunl 4| Q qulwlh F ii 0.4.IY4 HIV. ¢ 0 v. : ..-; ._ .u 0. Iwc v I r. #1.}. ....
'r040w00Ih . ¢ooe¢id4oofiwdw¢+1 . 000. n...... .._ Ir IJJ,_ . V441 A
_a. .¢.4 ... T 90. .0 O .0 . f o . o _ II...
1...... _. 3.5.4....Kwu urtt . _ . . 2-21....
- ....4. I... ..4. a.tA_Iwo+O_ vwbbwf¢.: - __,.chOA ,uuooou
I 9V...v »..Q.I.AIA..Tq|4 .. 4 4 _-v 7A.. «1. 4+ . . __.r a. 0 Yb u uoIn IA 9
r».>b.L.90. .Hlaoao+;‘ ..IHHIIIo» 6 v +-.. «Y
o... AcoQA .. .JQITA.ro,, , . +0. V} . I , v I.I-?.9- .
V .191 All... ”IO-YTALIJL. POLX'I". 4] p .. A. A I I.. A “I .- Ola-_lefu.
. owAr-o.:,>A_I+If.0._HL LE»... 9. .AHI ... .o A a... .. .
IIrlvqlbr .I. .+P.+I».;. ..o+..ooo ... 7.0.6.1... 4.... I. A m...¢lu _.
09-0-01. I IJIO “tr-ho. .r.ovr”o 0.0 YOIILIn . _.o I. 4.4-1.0....
IO Arlvllo v.0 . ..IAIA...Ow.A.« ...fi . ... 0.. 6.609 TI...Iv. A... ..4 rv++L
. u o.- .. 0. 4k.« +9? vi. ”1 HrrFLITvIO... ... ... . I. _OIOJlx L
I! oIJLIOr' FAA]! ..._.1_HIA .- fl q o a r (It 0.0+ ..Ir. . . ......OEIJ OI'IO «H
o 0 o o a o t 9 0.... r ...IO .1 r .. TI .I.. 0 O W....« 9 * 'If II. I..?rl kllzv .vIOIO. .. .Jv
o - «01.4. p... 0-4%.--.11 .. .. ..u.- A II..- 0.. .Oiv II.
a 0.0‘ r4 [fllul II. ”Ion”! I HWOaOL _.vu 0|. 9 Y
o P . nun .Firln Ililfit]- , . . 4
.I '4 O 71.- . ,. H A a ., . .. HI «Inf... "
o . o .... .. o 4‘90. I... 0.0ch.- o A 4 .0111
.0 L 7%. ... ... . . o _ . II I..-.. 10 -91-]..IOIO. 1+.v? fl... * ,. d. r. ._ I
A .. a .66... 96 u u _ u . . n __ 4 u 0 TA] r ,+.¢‘I.O-o.l Y; P," «L. . A ..A ._ o c .-_r ._ v. L r. F._.-o.-I1.1-I .
.H0 .914 I.... .._ n. .I.]. .. n p c u A. , a _./ ..I ..II.._IOIOI,.IIOIIJ'IAI‘JO.IVIOIQIO_IOIY*.I o4 _ p . IvL A. a a ..I40. _, O. .1 cl. .0 _.J. “.4....1 ’.1J.L
.. . n .. o a.» VI ..0 -...rOIzo , u ,u .. ... .;0.I+4-00l60|’ .VIL ”0.0%. JIOA 00%.: 00+ IJI. .. I».*- A. o. a ~Lrow —
H o o. n _ 0a. .... y I PO 9|? 0|. I' ... 0.. .A .o a o. A - . L . . . F a o o o a , L . -19.; .fl- . .1-.TY§. .I4I1u_ A. 1.. . 1| vlnl II. .15.... ..|_..II.IJ, »
«h _ I.L...+- ._w’uunro-A .. m... .0 Vol.7..- -1I+....tlul..u ..J+ . ._ c», v -+..m-.Ao o .... n. v .. *..rdu-d...._ '94... . . ...
«9L. TL.Q ..o+. L h. ‘ . . IOIYW. . .w . v. ,o hunk...o__.4r HI“ . uoo...~,..__ .AL. ., _ ..4HHWfi. . ... ...- ._. . ”,m. __ .._ .. u < 1..
0 ’fi +H.H.hh_ 00.041- H n 4 W4 ultb..1‘-l..., . ..:.IOIWIA‘JJA tl.....4lii1lu.ll l...l_.dlfi I.I..-» _ . ”_l$l.ujin.”00.‘pt u T .I .' u . ,. at I. ....F u If- I. . II I . I.I.A-HI...U. I.. . In- .,
1 0*47 ., .k A . A ._. ..-o_o 0 .¢ .. _ an , .. _._ c 4:: I“ I. ..,.O _V... .. 4 o ‘ . a . .0 c d... 9 AOL
ha. no _ a 4 -5 .r.... ..I. ...». 1. n n .1. QIFJLIGIAI1~ a.¢I¢_oIA-lou« vr. -...IAI--bI . II a on u . c I I.. . o. v.1 1 c .-..o .4.-«I.I...IvdI-V '
I 10 .OLIHnrJ. . . _. . _ . . . . .. q _ , . . . . . . . .. .. . . _ . . _ _ .._ , I. «I 1 .. - _ I I o ¢ a,
I L. At “...... H . . o b _ ., ., ., 0| A... .0.- I. (..IA
1. f.... . _ _. .I. .. .. ._., .
11 0.0440.“ A w? w , .AJT , . 1-....qu
... . ...: . .. _
Xv veto”- .A‘ v. . .....Qin . .. .\ ....u TVA
Iv... it. Iv.‘ v. to , . .. .T. ..lllvlr IT. . . ...H i ..q
by» ? b _ .... . . .II.IH.....IIH. . . 400
... ... L F 4.?» . _. _ ., . .+._J.A
NTHHW. 1.1 . .14.. 2
I010 a. . o o. A
“Ha. H. IOkl.l j 0, use no A
..H..'. ¢_L..I.O AT 0 0091. v.6
..-. , .4. o I....I I» I.. $9.1
_Irlll ..v.+l.uvu¢.x. . v I o .1011
a. LTA.Y. ... ... a .. III
.I. 11* L
E??? ALI—IF .0 r _.

 

 

 

 

 

 

 

1

ALL
111.]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

MICHIGAN STATE COLLEGE

 

 

 

 

 

 

 

 

II]... I I

.
v; ..

 

.19.¢

 

 

 

 

 

TILIIA'
W11-

 

 

 

 

 

 

   

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

    

 

    

 

 

 

 

 

 

 

 

 

 

 

 

 

 

      

 

 

 

 

 

 

 

 

      

 

     

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 
 

 

  

 

 

 

 

DEPAR TM ENT OF MATH EMATICI

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

IAA
.‘tt.4i-.ulil
. v
... q TLI
O A 9A O...9- .0 o. a 1
C. .7. 0 .I .. . 4 ' 6A. lo.......-H._U.I_’l OI—‘H ” u ..mL 1T
... 9.. . . . ._ .... . 4 4‘».IL0.2+ .‘Aa.?0..-.AA _ . - .fia u ."L .1
QAA _ . . nu ‘ A..o.!v_. . A , r; a too? X
0.. u .. u o. o .400 A_ . . . L .a‘ ,.I.r...§4
v.0 _- F ..AAO . .l-Ao ..oohl 4. .11 IAI' «I..- ..+.«. .4.: ..b O... LVZVA
0.1.. .. . a. c on .I..:A.,...-c.. . .H.v o..H_r..H.>a coH.fl 4,' 04.; LL
.. 4 -..T...O!.I'l.u.b Iain-..-... A. 4 .4. 4r . . u u “HQ
rlOIO _T.. ...I4IQIALI‘T,IQI. ..IuIcAIx‘V} .A‘. .. , . . .. ;._., .L..vfi9:..qax‘ar...01.‘9,f..wz Ito—IO. 1 HHIH
v ...... c......;.....H-..;c. . .0.. _.o vcflv.++o o1 vIAlpIOIAIOMA .L I.....‘w < 4
HO- 0.0”..- \.OIA..A+9. ...lo I. 9*0‘3 0.0- r4103... Y. ?;.oulcvv «ulah
I 0HI¢o4_....*|...44..o0ono¢¢Aoo» .... ...vo... . .. ......A4cw. o A.
IL+¢T:A¥.. 9.0.0.7... ...?v. I.....f'IQI 0“... 4c..Hoao . LM4.._¢~ “...—a v
00...;.....ooa..oc.to.oo.oooo ..v .4... ... 0 ¢ ._...oor are...
+— ......uo- .oo .0 ..:”...OaLOoA ~59 $0....o40vfo I ate... v.005O
ouukho, kn o.oo+o.4c¢_ A LAO. _. ...L ....».r.oo¢ 4L5 . __I00.IAJTOIOOO.-..
pooo+p+,a.._.oto ...... .0 9 01...!!‘90-10- .v,._4v..Y. .»A.w:1 1.4 an. N1 I.v.o!.¢.,«. DDDDD #44
olnioib+rrb+¥ Yfl'ttia 00¢.O.+0.I190 .. 90.41:. .4-4.4 J 0.1 AA Ann .r .1 ..-- 90.6 . , . . v0, 0..
Ir0.OOAFo_Io:...Aou $0.9.f4non -... «.0.-I.IQI-O. Io I...Q#__T.v.90o. >L .4._F.? 941w LT. , ++o .._ . ._ . .. .1 . .fi. . A . . Ono.
OIVI1.Q:ATO o .0 0 C o O .I. 9‘. fol... I.lf$0|9| I.I... I..Hlf¢IOI‘u ‘1.-..++OLI._LT . 0‘ 0.4.9.76 o t n 9 v . o u _m o a o a u o .. . . . . _ . w o u a H A
chcc-rLo¢+oo VTYoY,... .o o o?404.o¢-1.._4400. o.caoooo ......-. ._....“ .. .o m ._.. . .u.._u~,-. .... o Lv
-.II?01.0470004I.IA oooo+uo «0.7....4000c44‘... ¢¢.,A$t..<.“Lozoooo . o. u _.ur‘ on a ....u o to. - u.;~our to ._ooovr A
0......naoo. ._...u. ......oOo.4o-;o A; c . , _* oVo. u. .. .... ovo .aovvuth
-..IaeoVAa.;.ooL I. fri..- '97.... 014*”..u . 5 .0 L. .0. I -r ,3 I ..Iolv r.-.
§ ‘JIQOIQ..I!I+I,.-III-IAAII (Ia-1.1.9.9.... .OQQo .w+.o . . .- ¢¢4 -- a; A. v 0.0 T.
.Y-de‘ ..v004-o;fiIlIo - o a ... ._. c. _ c .. on o v. .0, .L . . . . . . . . . .c
w» .Hm.... ...,»n..u..m.*wm.nwo+_..1..Luhww. ..~.~-.. __n.-..__ , 9.. .
“— IOOI¥§Q¥IQA ++ Q 0‘96*- f90+9++1+9 o .... tI'r 101$. JvloloIIIH..I.H
I 0.0.51.9... _t .000... "ldO‘QO‘t I.. r? I. .0... Cl.-. «4!.
0 0.0.0.000rm.’ 0 Y1." ..4 ..oloAAH navio .oaoa
o I IDQQIOQAI_¢COO.~.[ L.‘ .74.. Liivo‘Qk. Aaoo...o._ .-
- .0. I £95 IL ._cIo o- [6.0+on ... 4lnrr+4 +0 fr a ,9 . a a .4 ._ u o
4. I ...v. Io_QI-§A *A.L"IOI. a... . 4.01 +4+I9¢OI4 ....g . _._ r
o 4 «44.19.- .. ...oa..._._., . o+ fi.m.co.4 oo. . ... .
0 ‘ o‘,ooo.0_009 waowoo r’AJI £9I+ArI .._-IOOIOOIH. 1.0.010 O'roergo.‘
0 ..ooofélln .Y’OIOIT+1V.¢IAI tMJIHJF 4...»;1 I o! Ac... 0..
4 .r ..741 In!“ 1 J44." 404444 A“? > Illl-II‘IIO: .I '04
> 6A-..+T¥IHHT11.HVc4. ..¢o_ o IL ...0056 ..a o..oa_ ._-
0vc¢our,.9 9......01110IAOJHL_»0 wAIOIYoOA ...o o . a...
)4. [[45, on‘av...o4tooa “IMO; 17 ADO.I.OI'.< .A:. .I. o- «‘0
v .0... +I4.»I.JIo.tIvA,.fIOdo o A. A”¢.o.oo¢ 0+2... 0....-
4 44.4 7.. o o.§Lol.4I0L4,u.. ’1‘...“ .o I... Io-«ob...¢.
0 Afr. o .._..— oA.Y1.oleI44I¢L ~4oooovo. A .owdo A.
w 0.4+.1 W.— F Hoo.wo-c.4 o. pOLIaJIOlr u ...301—u.
H ....o ¢,IT.OArI.vI¢I_#o.OoV¢A.‘~.9. ...u. __ .. ._. ...—......
. 3: ..m m: ...-..J 23nd. __ _ _L, I . -._...u-..-..-.,u. ._ . .._:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

‘

K

1213

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

      

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

MICHIGAN STATE COLLEGE

 

 

              
       

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

              

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

  

   

 

   

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

    

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

        

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4.14 4 4
MN ceava. IV
I. o..§.4..lo 4 U7
A 7% tot. _ A» v
IF: ...... .oOII 4 Ti
T. Oghy O O ‘1 0.. o I
J ,0 .o .n. . Toi... .IOLI
I .o 0-7+ 91-0 fin. All;
YA e ”+0 .
M9. :3
. A
D 0"... VI“, v
IIIL. o¢r.9~ooorvo
4 _. r I;
I‘ .
> 0. .
._..L . .... ...) .
Iv ' +; J .41 . . . 041.1%;
0 o . ,1 .....ijlnlhuv‘ Y1
» _ L!
.c . H u
17. O fl “If: .lTOIr . . 01x. .. I
4 .o‘llp xiii .I.“ 0 IL K s
1 609k Quitei 0) oJI - ...OIJ H. 9.7.0.8. o
Tnxll*?§. 'II, - lifioio+1LtIvO IXY4
b Dirk . P¢K1Io. 'TxIIOII..§QQA O
« Fl- I.....v a 77.3? 0 Aw I
LII—ya L7.» 1 .1. _o. _.vI..r {r
LII v.0 .T§A F . ._L_IT.1..I;qu..|IuIO.rIOUO LA IA.
17¢ .. ...... ...,.LL.I%..LI-1I.LI9 IH
(.11 Ut..I§II+,[ .
II Oa40 ' ..u . . I.I.HU t1
1 0 a o ‘.I1Id.i.u?.u.t il .0. I.I! II¢* 1 II . f ..1
II 0 o w O u #LJP . ill-HQ"... I‘Olo‘iquI'
III$O¢Aoo .r ... ,. ., , 4.9.1.;
b — a _.OIQA , to. L
’HAVOQOOOA O.tqt+o |c|A0.uPL
II to». . fl .7 4 .J”. A .0...
10, t. 0.», ulfi 9H «#4 NIH“ T4; , ...
Y ...... 101 I ,WINIMIAs1. 4‘&
4 1 < I I4..4 ‘
v f'd‘ OI Yfi «no .19 . __.wJ.o .7 lHluIELAw
fibWOOTOOQIA #. Gunfl T. 7+...”LL-I1A 0605- 4
Iv .VIII *6,!.III . _o o u «A 04 1 M H
vultnv _VQ 'T‘I,IOIQDIAI II I o o to. p c .c .o u 6
yr... _0 VIJlflorOOII . o ..I. w.
T: OIOI TAIO uch 5
IVJOoIfiIflwév vi OI?
VAvflIWI I 17+! ,1. . o b
Irflfllfil‘it, JIWI' .It . %
wfififiVLI'IIJO
Iv 1A.... .. “I.. M}. . . 0
Y. .o .09.... a
I H.027; I
I I._I4 .I
I . o _ .
v i... I . ._. .I.
1?. o o . u o I
I... .. n _r _u . . ...
v.6,(c o, I , 1'-“ OOI, .Ir, , H ..., I
I.I.IIII I! .,~ .4— ”IIH ..I.II ‘I ."l .I F . . I ll- {4&44
I.I- _ g.‘ ...: . .J. .9” t. on J " II.“A 0.1T.
v p ‘0 .. . .— _, ..tt. v .70.?OL 00.17
on ,u. o. .t [._IoI .._. .66.;vIT
v u w ._. c ._,I .._ .LTrI
I. . IIl.... t . WA [LII
13]. .I.! I 1.
wIIoo.6_ .a .n M
....Qo ... o 93%..
1141i _. a III
.I. . ...I
Y 1 4. . .0 9
. .. .0 ‘ A u I. .VA
v..-._._ 0 ....Jd‘fl004¢0¢. v4
...... ._., IL._I._ I c6
‘0. o u. o.
r.-. ._. . . u ._ YO .v
t . _._ ..n .. .._
#7.. u .;.u fl_ a”
O on ..r. I .F I
v. '4‘}. «H H #1.. fl .. 5:“. W. . 1v" , lulolgfil A I
.“rc.. _v o. _ o coo.u.c..r..u ......
I01. vI.I.,Ql.- GA lab» I '0 f2». ......19... Q..‘_ H I
¢.L.rnI;. ,. o o. ..roc.¢<ov., 0o..+ H I
...I; OAT-I .F .ti’u0I0h}.rn .4. .1__. o. 0..
I _m ._ I“ .. .....v....o..?._l O'HvHL ~.. n
.....Yvév‘lbcdflll .... . Tataaa..—_.r .— v _ OI¢¥_A .I.“- V
I Loo«.«_ , .onovo. ofIIIOY PA II 9.04 9. HI-
Iv’ Io.o- m ...._+¢OIvIbovo+l ._.» DI 10Lu4§6 1 .a ”fit
bL. . __ L. r. bH. . _—
1 ’90 I < 1‘ CJ'I 1 O.“
IrQO..QICIO _«LI , . ..IH5I+.C t; {I.I. .I. *1. _ Or: _ w
IYTIo.o.0I9_P.r . I ..w ‘o.. ... . II «411 (I w
1..., o . c. . r to... a k*v.,. f I Cffl o 0 ET! . H A;
I: . .04. ...TfooOO +3. .901 . . ._....4 9+4: +. .17
I ThrfiuVO. YLIO H.«+'IQO .’ . ...on _o- . n . ”o A;
$190.! . .aaao} bI+cI1 .. :. .... . .;
III»..¢.09IPO ....ao... . f . b * JVIYO 17760621.; :%
«+0.01; . ._. ._. o u o v . {0|qu .IOIOAVO b
I ~oooo. ,Iéba I... o _.. Y”. o‘ofl.vw 1
.¥r b V i
ITO“ Mo 01 .0 YIIIOIL .1“ tH‘ M m o v
1:: ..HR r05$+o_.$..1! 91. w T... H r: .4 o
1Y9. oI IH .auaoco- . .. vii ..4rov4
1:..caoav bw>.?9?.v .. . .... «+a :1
1702.16 .4 oviIo.‘.+I.o.+- 4“ ._u . ca .H . .>>
11.??oior1. 3.509.... n_ .1 _. *++ _. o . o. I
Iv 917.10. .-.”.OIH TAI .I. .I. 1 0 +4. Fl . u r w I b . . .r I D l I 1-
LVOOIOIOIO to JHQ¢0+O+Q a? ..II... 19HIC+IVIOXID l¢ d
It. Qo§$.o_u..I IA 9 o o u a who . o) . 7+ » «IUIO ,. 101 « NH 0-
‘0 0» J4 ha» b .I.: TOO-.... “I. b 1 b
1.09.00.9Ta. .Hooo. ov I.. 7+1: ,. L: o“... ..L
Lv¢¢¢oou... F A. pno.ow. .... 9.90.0.0; .._. #1
...-0o. . 04. 4Ifl01_0 ... I'lvliu5+ .14» +r4r
YVOOOvOa . Ob. VIJDvoonw v‘ -. ..IIOL4 vw.¢v”,l'fIfidl. .
Iy?o0ro .H. .54 T<I....ooI0Io 0.. o. ... . £016.» .1 4. La .0.
Ilrr.l+lv.bb .o r_ 0...... I... V 4va .14
IJrQOOOOHfiA. O. I .III b.rII fine .__ .90
L onovur Io... WHMHYW'HTOJ . h am .‘
I I ..Ih..I [.II L ...
+ 1 j a _ . h. . J
17¢ v .vum +u H ItI. L , L [14. . rt, . 0
A *W ...—OIH‘aOh o .w .._...H t;” in.
-‘IA _.4'CW.’. I C. II_’. 0 .’
4 L 00.11.0101 ._uIc vq Vio 49.v
h“. .bOGIIOIO. +9,“ .._. o u A w o.
o... oI7+YvJIA ... . To. .. .9 '9 . . . .. w t 0
F030; A.» 'caIci4 oa+vo. vo‘Y. I .«+. v... . . LI
#0. ca; ...: .flo$.¢v5+- 19h. .... .. . 1.11.: IYrIIv yr
IIY Q.~r ..t .91». .I.. ..+._+.. . o .. o. OJIXYi
4 '0 4 4 51*]. 0.1901 ‘1‘] 9.4.4.171. { {
9.»..11 ‘63.. u . . oIOrapII .IfQIl ..q- JqllflbufrwJ. .. .I. o ._ r... v . ¢ ¢ ,. c. :r
1%... . v. . ..IOIA .IfisIoIYIIu... ._.»... c... ...A;1
OIOIATQI . v s o IOIIQIO.O# . a . . o m. . cl I¢I+JI0I a v. OOQAIII
11.1 vo+gf Vole... ..avm o.-ov... ... vo.I¢rtJI&.T?o .H . v 090L111
4 .1 vr+yY0I _ , . 00.5. o _. .0...-" .Ho «.94.YIX
I ‘ I c or... 9 «at . ~ . .. IA .,_o.009oo0-4.- 1 . . 4L1 I
In 0 I 1 do Tr]... "*th ___ .._-YOq’uIIui . . J
- * .w .. . b. 3;: ._. {1-2; I...-IjiJ . .
u r I.” :73»: ”VIII!“ ..n H

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

DEPARTM ENT OF MATH EMATIC.

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

DEPAR'I M EN T 0' MATHEMATIC.

 

 

 

 

 

 

 

 

 

 

 

 

     

    

 

 

 

 

 

 

 

        
        

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

MICHIGAN STATE COLLEGE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

    

 

 

 

 

 

 

 

 

 

 

 

 

 
 
 

 

 

 

 

 

 

 

     

 

 

 

 

 

 

 

 

 

 

 

 

 

      

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

    

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I44 11 I {I41
4 444444 141M HH4fi 4 . “Nd. AWN“ H99...

LVS.M.TT..I9. 1.9..- “.._..- T. . . . .-:. .4.-..
l IoObIPOOLIorI-o . ,9910900-..’ . 44 _4I0 .99“... u — .... _- r-. o o.
I oLI..VIhO....IOII, .111IIooo v.04...oo vvoo .0.. Y. _ .... F. _o I
I ”I94. .%;oan . _. ..I-I..... 4vH-0_Hrr .v+4.'0..+l._.‘Io , I. I ‘I ..I_HII .0 L
Ivl.10|6 TO‘IQ'OIIOVJI .. . .:...-0.... w... «fl bvvvlwlv4fi9 ..I... d. .._ u.”1.v._otI
nI~O.OOII4O,’_IaO-.o«o . . .000. o vwoo .wvforo h ITQIdI IL.._-I“-VI“+.I.>OO.QI904
r1 .0... 90-09 ..O‘OIO.OOO o. o. w ._u' 0. O I+ ...I 1...... I.Iv.'m_. 0.
IIfiY Cotonuvvoc k.k.OiI|Ia.+I-AI~|.1vT+fi .I.-it. f4.» .rLOvIr +FAJ.I0..~.I._. II.

I ILI. .... 2.293.; 3 . _. . .L . I....KLI

I» I O ...-9"01 "Md—.4.... 4 (Ia-.1 I ‘11. ..l.0l'.'n.!

I I.I to. ._ o... .... I I . ..”v o ._..o vm.o+anaw. m.»..w .:...?OII
lu.o..4.. Id.‘.__ . ..[IoIII-I. ob o. o; “..9654... +0498; . vooa. _ ._..o-...

II #¢.¢..w._o a.¢. . u-aoovovcy 06.44.“ I .0907‘ “......o.o_.ooum ......boé
IoIlIrIJrIrIII'd VILIrV ..IOIOOOIO v4 00". 0“... ...—r ......r..._, _r n.+.6....0 _
IIY.'FIIIVJIOOI§I0I0.0.V~ I. 0..»oo .. 40‘904- o”.*ooo‘r $.09o*..u ..< .59.... I

90.0..Ito pro..-ofiuo.. o. o ....o. . .4....' .....u.. «..__ v. ., v

171T... ov,+ . 9 .._.. . o .00 AT. +I¢Lr¢om .02.. 6‘1I0..‘”_._ Atoafis ._.ffina curvo Io
lIv-fiiwoo vnno to... IIIIOIKI i. 0.1.1334 IIIIJ 6.10.1049. :41 O: .4.-II 71+? o 0.. .

IYAIVOA o a»? W~fi0+___ v ... I. 0.11; I..I...1I. Iolo‘IOIOIOIOI.(_,._.I II....II|¢|9II .10. PI Hull |

04 441 L4 . 4 .I -III-GII-IJT ‘4»{1}; .:.oI _ ... o

IIIYAV «4.. cur2~Iw . v ...v o c o . » .0.-.... __H_ In...1ro4~_ Lu‘ H3

Y.,¢H.”.m__tr.+ 9 7...... .. 9. ..........4......4.. . o... ... ...

fiat... o-I4I6&0I.rLIO,OJ_OOIOIo.I .V ..II QI,.DI.VIQ.I..—:”Oo....too r 00.“ d,._u..r

., on“... T..;v+o “voo.-qr c. .004 Io‘c-Qooow4 00+000-0. TOO vfifrhl'fiI

Hk.rr,_l ot_.+....a..qI.I-.vw_cI.-fo 1. Trb. 6'3 III.—o»vor. _oo * ooe ..I-.0...

IVYav-Lv H 05.... o. o; wmfl.o.. o .era vvovbva co. 1...... 4.09‘ nhv u ..I

4. o+.p .fi.oo..t ‘nrwr a _— v00 9 ...-cw T»4fi.0c.Lo-..$IWOIO.O ..o I 09

II I..).OIOIOI I..... - II1I+4+._ lol.‘ 6 .9 O-lblvb Wu 000v~oo ¢F~04 4...-00'
A ..vo..¢ouu_.. ._. ._ c" . Mg... 0%....00. »_v.oo.oo ..-oow 4 441. 4

“#1 0 ll 4 44 Ylefi 1 A I_ H A4 # 4|... a1 7 «o; . .«
.:::. _ . .. T ..:,T. ..II-R..- I -... : ..:-N.
YIIOvboo? bond-I.I...i 0+... .0 14 LIOIIIIIIAVIfi 4 H 44. a Li Iv I 09. ”_.v¢

OI I..-..Ivo. .u o ...; OvnI. - v A. ciao _.Qoofirl .14. II4F4 4. Flo-III 0448.- . . - . . . . . . . . . .II ..IY.$...-.._ HI

.I. 9|... ChuTI I oIvIoI-v 9 I b; .ILIo c. ..I'.+ O... .Lfl ..I . III? 0 9f 0 WI? 4 I - .5 _ _I._IH1J|§ I.I. ., . . . ... , . . - . . .. .- . . . . . .. ., . .+ v 0 fl 0.. _ _ .

..r 9 .. ......o u _ ._ .. __._a .:.... .o .v o».. . «...I..II-41I,Io+1cco|91.. , . .. . . . _ -_ .. ,. E 3....

0 o 1. o.- - I v.- - . . . . ”AI til, I. VIII+ .....IOVIOOIVIVL'.I..’1. ulIHJ 0.1. II_',-.vII051.- . . . _ . . . . . . .7: I .T- _ -.

v u . _ . . a o . _ . f _ . -+§4 CI. 1:? '10 ...o YW. - l1 AvlnlI _ _ - ' -.IfiI. . . 2 , {9 mil... H u c ”HUI I.I.c .I . .

v o o . . . )1?‘..I¢IQO v.15 Ifi4I 4 . w. § .JIIoIvaAIoIIo . ..kI . I.. H _ a .-

c WW L a v o ..I. .0 a (II .0....x. {I0 0 .v JJI IIoIb. “A“ ALI.“ “I 7‘ Fit...” U 91 ,. ,. .. . . . . A, . ... , .. -. 4 .I¢ . r HIM.

r FOLIO '0. no. .I.. FIVOII I y 9 J“ 1., Jlr‘l‘lfaflfiglnIOJI- T . . . .IOF-Iul tlr.‘

_wII. «IV» _0 Iv,|.. .....OL1051 or .I.] J OI o u‘sfilw_“¢- u. _ _._._ ... .o A
"Ml- _oJ- _ o u o ._ _N. o a c .1 _ W. o . to «do. .vJFOXI‘ o c _ . IJIO‘IIIIOIOIJUIIIOI .O ...II A
11.9.. OQIHOIQ'YO I. ‘M,_Ur4aovwi I’VO'OOOD G no . a. Cote-t.I-..,O..IIOI4-CII¢IO
YO. a. o 0000 T4.9Ib_'# v I wart. I n+3 n.r_-_. tho-no.5. u... ‘

YI_..IoI ..I4I. .4....w 6+46ch>~ ._.... . ._.. .... .mr....“.

b _n I. _ t a. _ ..IL.‘ ..I-_. Ot.‘.¢l IIIIOLI¢L2‘I.I'I ..I .- .CI I.I .I. V o... TIIPJIO o _ I. H r
v o . - 215.. 9Lv1v_4l.9 .4_6.9I4I44._-II.II . {..I 9 In. Ia. II\_l oooolI..

Y. a 4.. b _.no.__ _v u .o o 1... 04. .__..I, 2. ._,¢0.o « -_, 61
I _ _

0.0”” J_I ..-._....“. v.~ _IW. ... ___. ..4... . I. .I.—._t . _._._ _..... l
19.90 .0. ..I. . ._L.I..04I_.OI¢ 31+ IIu-I II.-_HOHI.II. .. I... _lhlu..l..urfl lfllwnathfns-qwih .
YO. .- I. n. __.- .. ... o 9volo . o o c . .4 ......II I
I. not. ~I“ ., to .00 ... I: In. I I III.._ «..II? I...¢.-IIHIIIIAI I

I , I. _ 9 _I,- I .l» ___-.56 ,-o .0 tr. _rr - n
.11 I . I II it -_ $4.171. .. IoIoJIyloI, I .._I J... .. .IJL IAILI.0IoI-Iu ..II- om I.II .. .- .- . fl
r o. _ . _a c .r ....I 0.... .L- n .A» to¢_.r.c .0. v.
vsaI'IOH In. 0 Ola ~I-O> I. -uI.__ u b -O.' I. bu. “00‘. n. .0. 90 o- I
If..o._.v .. v..u 1P . w v _.a+o ...... I d__ ..10v00.v “.4 ch _.-¢.4 v

. _ _ .I ......OLJo. ..II laI _I._III . I I: ...-v. Y6 _t .I. _+h _u1, ....

"H“.OIDI _rlu‘uL-FELIQGIILIALIIW IIF-ILIn‘v 4n._ 1......1 .L1*/‘.. #9... ......” A

.I . I ..._ _ I- I I _. .-

I 4 4 0 _W4 11% 4 lat: ._.I‘JlfiaJ 141.1?! .1!le ILIW ['4 .— 4. _ . . .

.I I I .... 0 «IT.» 0.. OIrhilIlull I..-.._...” u 614le OJ .DOIVIIOIH 1 . _A f 9.... .o CIT. _ n . _ .. o _ .

. _. o. - oorcvvao. ._. to."rocfic14d«_H_ _4H41w 4.. .¢ ..I. .._-__.k 9 o A IO;
u E Y‘a”.._._om', .._r .....n. o... 0. 97044.. 4 a. a .I.... I n.0ioaIO. 4.,+... . ‘40 VA. A u”
I o I Iv- I14. «I?» o.“ .410 . +4. fOIo..,. , ‘IIATqIf.+uh iTT .YQ T‘Iflui a" I I. .ID.Hu5.+1hl H...1I'_..1.fl IHIHO 4 4

. h. o. ... ... ... . cvo o... ‘9 A .v..... I440401..4 . , A .
In M I I o 0 ..HHAI1IHHIV-Q T“ v” ;Hfi .4II4 W a. I.I.OIOIVI + $.91“ c... 1 ¢ . t. ._ l_.I.-U-IOI0....6_ .FI...IHI¢I ¢IIJO OH O O _ H.” I... C O O 1 v41
1 9+. ...99..:9+..-.I.I -4 .419 -94.. 341...... 1 ._. .:...-..II J3... . f I i .98
I Cajun? 4.-.;14104IqongI HI 4 4 H .4. 44er 4IAI .+.a. ..HIWIQIToTfi. I..-.I, at... u.r.I..Io acoIIon . IIIAY. A9

._ I.- I...- -. _ .14 .1l 41 110 I I . a v I .9 9 0 I . o A .

J I. F HIFJ‘» HOMN._-_ v I ._.IFI I 4 I .> » fi_ H H n r+4+:: ...HI.I_ 4 _ _ » I 4 44 «4

c. _ 4 4 . . 4 2 . _
T. o ,o ... .. fl.-.vf¢.u ~.H4-fiu.w .9. 440‘. .o .0.“. .5r Ttk-.w_oII6...LflJM.fi.HH ..o «In..‘-4 vIW
I o >- -o I“ v._..-_ ,,,I omoo4+H «.... F...L..oa.... r... coo”. A .IHF.I «4 «_.IF.+..« . ... _.H_. I
I 9 . 4- 9.. 7: Hw..u....w._ .fi...;o4~:o.o. . o.L .... ..w H wLH4 fl4u4 4H4. 4.”. Y . ... . , . _.-*,_.E u

t .....V. ...‘o _. .F. .. oo. o A: 7.6hv.04 I I4 «I 4.4». 4,. . . ...-..I .I.l. I._,.. I u
H H ”H qw- .Hnoootfi~,wo.._+.fio.¢Lfi4+oo. o 1 o...9+.LYOIraIO+HI.w...IIIIIJ+4LTHY . r64.HIHHIH.+-.ohd\. ....v.—wtc+.¢o.-vc.v...o_.l.ru 1+ WM.H. I: r.
1 o .H is .IrIIAT07635oo-po +140, +Io. : 04-» '1in I 44‘4. ... ..”+....r.H I-v. II4IH¢OJ..I 9. I.I-ILIIII I Iv.¢._. _. V.t.4> I
.. .9. .r. u.¢.o_.¢+ c _ a... v9.0?»HrIlrr6I1IoIoi.rOIoIoIoT.roo .. o ..o I.. .0.... . 7.4.9, .74 L _ 4

n .H oh,4 2.. . ..csw.u ._..T ..or> I 44_ I 41+...I._IH..Iol- yet-_fl. .. ot.+-.o-.-. ....I..o.-,Y.e.-, .-Huv.M>
I? .hlpu,..4._.h WI owoaco..n, «I. no.» . wow. 0 4w 91ft??? Iv. 4......“ ..o HI. I

44.44.“ . I 4 4 4 4944 4 ...~ .. ...1...,1Io4 YA

u OOotfi._.wd.4HM .... #6 vmoqv..u1. “HM. fi'bw r IIW 4 II ofiMAoounauo> ..w... um“ » I.....oL..FI.oI42-. . .Ifi‘oxwr n1”
. to a 0‘ ‘1 u. ¢~ O ...000: - 9‘ AITIDI 1 4.4 .4“ O C. . n I ,_. I _ I..... .I. ..

Y. OOIHHIOuHWIi ..IrIaIrIw _0I1'Jw Ia. .._rlleJI I I.I. 0&1 QJHI.a 0 0w 0 eATo I M. .9 TIOIL. t 14.648 + .o w .. Ho . w _ a a u o a." H [a ._. #1. J_,lI-<. o r.” - - .. I.I“...lwn .Inu

Y§I1¢I oOIoI _.I~I...I_uo_..-..I.o .ALTrHOOII6.oe yIHeo a." 0H7. LA ¥ .rL—I OIJ+L Io 1r... <fl1 one. or I.....“FIL v .‘ ........ ......u.c_¢ooo4r..T.I

rOIHInO IOII ....A .-O_ ‘g 00 T1; w IIIIQIQIAIO &+.__Oo. ‘ v‘§.YAr .4 4H5..LIII_oLflw.din fiQkQA.4CI . _F_o- r.._ “Pi-0.0; oIfLFrfiA _o|_0.4l.ol_I:l..rQI-I«. It; Tl

..a .eooo ......ow .- AH m,,99.00.os++dloof1JHCOI4LvOI4Lc hv¢v.. . AoururVOO Ifloevu I0. I o e.- .o-_.1 .- r.tL-I|¢I_-.wl.. WHfifoTQ... 4
v.60 ..IOIOdIOII on A...’YI.. n. ......oo. 4a.+.QVV_I 4 I 44 . $1414 I .I 0 4 4 fi» .0 ._I_. .. _ ... l.... In-._.l_.. PIIPII>
v 3+“-v, .....OIIIIWI my. .0 . __.~.O.OO.IIoIrI-#OHH.4!I04 44 «(011“ +¢f904¢o 44H..96?961Y4.0_¢ «4.9; H .9” a “or. H .Iurallm..v YHHIIHIIIHIOHITHHOthrta HA I.
Y. +0 ‘ 0+ 0. 00 ._ O o . ... m _JO .0 VIC. .AYFIOI. .OuO GAY...‘ v 0 1 o v. r o b D 5.}. ‘Ir'I’qu I j . .. uni x o o c ,_ , .. 4+ . _ . I

It I. Hr II I I_ 4I I r . h F » _»p p I In». wl4 LI AYOJHIHO I..-II 3w .. t I. I ..II IQtL . Irlol ...wlli‘o I11 I .I....H. H
4. 44 4 4 44 14 4 4 4 . . . I I .. ..I .-
4~ {LIIIIII I.Iav- .I 01% .4vte6._Ion,IlH MM “0%“. 4I IA “on. ~eco¢~ . . . ..I. .I+I..m|. _ n...+rcI\II9ioI\.olI-o[9o no It... I
O O LIJ."O.IIOIOI.....:IO .. .3. Q . o o o o 0 at. _t c 0.9.0IWIHI'L'I THIO Iv I _ _lr T§._'IOIO.HIQA 0 . . . Fl. .’ IF 0 o t (\ _I.IO 'I.IIII.’1 1.. . ’4. a o H .fi
”Av 04... ...coo. . ..- I4A .iL III-...aIOIaoa A. . cook.. +r@.. VAfiI*... ..uvo. o. ...++I«+4‘o.rd 4 ofieoe -.¢ A
"3.17%....” ......o. .. 1.3.0" 650114-...T4 «..-.jo..vo.*_.L, ......IA. ALI-I.. ......» . ....fi . 1 IL“.
¢ 0. ...... ,n . v.94... .... o .04. o. Vt.va.6o.6. .OIOIoi. 0 . .... ._ . _. . Fm. . r
III .I"..Io_unrun..?..o-?o?..x . » 4...-»190IIIW- 4. +I.I?ovv.ooc+ ‘ ...... H. I«-.Io 1 . ..nl» “HI III "Infrv .nI I41
“9.. . . .: 4:9... 91:. ._.-._.?“
I.. u .. 9 0. 9 O. I I. '1 ti. .9 7 O 0.0I6I+ o T. II» > _. . . _ _ E I
. JOIIIt-O.O¢r. O I. a. 00 4 “ HI ¢ e o .n ..I. .IOI.l .. n . v.1... a ooInvl, I 4
_ Ir. ’0 oIddo o ......o o. 0+ __ _ _. I} I r I
I: H Howm .H. _ m... .. ul Y0I+u.._ yr ¢4Iy _ r b r W- «r r I .1II.J~II6- £6191... ..I-_.TWIPI .4» 4 « auqo4 .L T;
_ 1 4 4 4 4 4 ._. I l . 4 9 Jr ..I-II I .. .1 I ... v.61 . . . .
IVArLA‘A ..N)_‘au ..OH .0.“ no a + v W’s. lelmvf I-TJ-+.O¢ — OI "IOHLYIWi-fiwto _+ 'ceo #0 I u _.w—- .000 I1."..JIOT+I'¢I§JI¢I.INII<.IQ, obva#~ r
Ivtfi . vac-....49.o¢ .- H I. 4.. 910I4¢JHIH o .... a. L1H...‘ JOHW OH-.. I. H _IoI I.Idlo .YIF. To.¢.¢i;.,+d+.¢o o c. a . I
«ICAVIOOI4* 9.4. (I. IT“ . FA go. 4.... v.00...0+¢ A ..IIOIOIO _ III¢4_4 fl 4.. . 0 K h o ......vvooov ooomoofi¢ ‘ .ALLI
Illvéooofwouo. y... u... o... __I I! IA.O.'.00IO¢O.4IIJ IQ.9¢O¢+r ...-cfiw_ 0%.... 9.; 11.0 IaIui I _ to. yo 0000‘4_.cvo$Afi-._ «VIII-Al
III... *ttTIToo Aoovo.o .. .ro 94¢,‘H 01.0.IOIoIo¢I.OLIO.1IvII *4. On 09 a. f v.9... vvuoI I -Ir+¢ooI ".:...... oo...o.mv +1 fI?»
{Pr#+Tbivprbofi¢Iv.r I WT.’ .4;4.'f.146¢L+4IT0¢0906 .~ ..r couIHv.u.+O+ID Ir . Jor— ..r.. . ._rrz.~L.1.Ib .Lbolwv. 1 I»
H» ..OIOLIO vb. o.- rHII .. I.. ..-A, IL... .I.-«‘o ...... .. 99. IL ..feOIOIOI 0.. . . _ 1H . H».
1T.IJ+ ¢ ._ Fa _ a L H co re cYLIAOoI9uo 1750* 1.04. L a 4
H>o.o.&.+ JooIftIIO.Io-_.9-.. 9 04 4444? _w o. .4 Hoco..WEo Iofluoow... .10.... H _ 4 o ,. _. _-_ LII+o _. JIr “VIM. . _4 .
. _ . .l.i .I - , L0II4 44 . ‘1‘ d o ‘If . .
MN“ ..I III... .. r I. ¢I4MIWI9 0H. .r. 4o“ ._ rn.w.o . v . o .H «_I LII _II H_ L. _FoIIkLHOIIOIriIooIAI VII...»- 0» .Ter Pb” »
4. _ 4 .33511417 I. .I. - .. .441 _ . ._ H . 4 H 11H: . 4 U. . _ .. _ .¢,. _ II-.--.-.{.....4_.4-_4. Lifw . 1
LI. vOa‘IOFQTvIOaCOJo:. ,,.o.17 o. ._ #99:. ‘40OOO,WIAJTW1L H.H.14 II] .4 ifio‘ J... n. o. 0.. u av! IthIIfir . -.-u.o ;_. envv v04 4
IIIY?.Q.+.-0¢Irvowoz.oo a o¢ . v 0;. 4.6.1... vgwlo9fb0 A1: . ’46.. Yer.;o4.4. _._...‘4. ..owvoo_ TooOvI. D o“. . ..ooooooo ._po. .0.... tub “99le ..I I.Itr
IILv0¢~a.., ...-om"... ” .._-o ¢ rnA...4._II4 I .06I6d011'.+l¢~+044.4 o .Eu. 0.....I-4. .u 010.00.? alIo I... r .. a. .fi.... . p I Lu¢ .4 . tor-OI ...L III 4 u
VIIYAVQO 0 . «enroau . .... v > r. by '9. +I010I3171I04wtfi¢040 v. 4..-— H— A++oooo :dq'OI'I._u¢Y¢.OOO I II 90. n _o. o Do u; L” 00 VIIT b 4
II o oo. .v-orT.,?fi-+fi»OOF|¢! H > I4 T 1'60IO. .evvflv.....o» A..IOQIT.VA-;I96I.T0IIOI46 I.I-_.. ‘;a ct. A . . o m. o... .-_. _ 1.. I.. 1L...IIL. _«_ 4_I44 II A I
I.I-v01 It.Ioo 0.4;”..4; to. _ .0 o.Hoo a... .4 ..,Av...~o-o. 04.5.!46: ##vI-ooo . ...... ao’.p‘_. ...vo._- m I... 7 III .1. _ l. A ..I 9“. ...4*.H «.... 9. III
IIIVQY¢I~1 ._....oo—o. .4.. ... ova» I01 TIgoruq. .o-1YT.-.r..l OIArdedoo. 1,. o. . c a I. _. -o.o . I g I.....” I ._. _ ._...”_. 9. a 4.4;. . ...... . .0... o.?...
Tvvvlv .... .., . IHJ. ou.fioiui VOTIIHIY. ¢ITI Io nw4q4c. ..oooaofl.o_?o o ..II. 9fT~HIIAr+II¢ 744;. "It: 4.. ‘. .1. .9. .._ o. . o .IvIIIIe. I6 «I t Iv..IlIl
LAILY HOIVbLaLA I41“... 4.4«1 _ ‘64640WA1YA 4 no... v+u994oIOc¢na I.I.» 44 4 Ifi 9 IIIW.04.OH».~.. wH.» map ..upo-deJIIoIAK u. .._n L 4.. .
_ u. . » n L . n u n r F . . L. b r- 4. .J
9. no .L I. _. _ . H . I . W .. J1 .4 . - o .I I- . .9 I»
. ...... . {Nu-.I.; I ._ L a . 1113.91.34qu LI 1.: .UHJZHUII ”I.I«k 2.39.2.1 I-HHIIIIJMIH? I .- . ..II.
rt. '5 .. coo‘ua .I o . ....4 ¢ ... u. M .o , DWI. ._ -. LL
- :2? ...-.::.. 3:..: __._U; 2 .. .12. . . .2. ...”.H.“ 33.33. ....
_ _ V. .L . , v.95.tI‘.U.OIoI OIOI. 00 t n .9 I IQI‘I — - I4 s o .- . o H o I .7 _ .
l VQHOIA...‘ C cflfloHOhI - . '0ATOIO.Q-Il-I .II‘IrIITIOIIOtIO 04 o c 9.000 . v LII OI! .__... I. I-ObthObOOto‘iAA $1 I
I 4I I.I ... IO. ., u . .
> . LIL‘. 19......¢._ . 0.1...9100.In....¢.I0...4..YIIvAI 90¢ ll_l a A-_ ... flo.ao"ro L24. «._I44 40' ....v o.L .0 00".”. $ Av. Anmu
. 4“. ... cut ..I. . .. 9 iIrI§ ...-0‘... ......” .o ur._I at I .... _.14... _.4—. . ..er... .4000 o IMIQFA: ..
. J l ._ . . _ - v . I.I I _ _ . o o o . I 4 IIOQIOCI 1 .40 .41I4 L4
LIIOJIfiollIOOI. to . .....H..w ..nnoIHIAoIara...YI.IH4....v..uHIMI.o“.._H.n J". IHIIIIfra O..o«.l._fOIo.ln...lIo _. ......I ...4.. o..¢.‘.H a 1
I v ? l-I..II<I ..I-.11 C 6 a ‘-A VII ‘4Lhr, v n A. A .I.. I! L 4 r _ I ._ T ... 1 1 1 ‘ I I.‘ I. I4 .—
I ..II lull... 0|. I110.-”ll0104i1 I'll-Ollfo r TOIfiLJI-iniaw I a...- . ; {II_4 4.. _ . VI“ A I
_Jo ..I; oH-owoo. moo-,QOU-Iéooo ...mnoo.-1I+:.I£.. .HL_ a. m. .""."H .ocnnuuno 00”.... .v
I III 213...... 41-4....I.”..9...._.~_.q-.......-..x , _....I.-,._. .u ..:-..:... .... .::-.. r.
YIIv Imu.w._oo-HHWL FIIuhIHflTcfltuunvooII. ..4“..oo_._..—%. o .._.OIIvlnoa...‘ .. ...... c¢¢«7..oo.v9. .11YII
_ _ _ . V I .9 4. I _.
.._ a. 7.1.179. .140II.OIT.I4..T4#1¢I¢ .._. ‘1. ... o o .. a. . u . ,+_. . _-4 _. o ._ «I... , 1 . . A. I. .4... _ ..I a +o&.o ’0 A at!
_ 0 IQIQI+QI.§4..¢I+_ !.II.IQI¢I.6.. ‘I<I 11.0 «.0.. v.0 ¢VolfivootIOL u p ;"c._vt_ Iq0.v OIL... I910 I¢ 4xfl0‘44,1
..rwyfimflnnwn act I. 4_—_ . ...... g. ..E“._ .. ._. Tic]? ..I” . .. o. 0.9. ... ...“; o.._.u._ nuo*o.%
. _ ._. II. ..I. . ___ _ A
,5 _oIII1cfi 0I.~H1I.. o. o. ”.9 69“-”..lerfih....._. .. nu w.n$....i_,4rml.u;¢..«IHILLr:.. , .w.nun¢n. . “.....o~u4 us» an
.r .. ... 1.3.333 WWII... . .HI.I»I u- .I ... I F: L . HI. .I L. If NE. _ II I ::

 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

HJ—l

I

J—O-O—t 1-
o o

 

rmn

TIII‘II

 

91

 

 

 

[11111

L

IIIIHLI

I

 

 

 

11

 

 

 

I I

v

 

 

 

 

 

IJIIUII

 

 

I HIIII'

 

 

 

 

 

H,-.§«oo—o—0oo
' b

n
_
C
vol
ITI.‘O ....
a I. C
O c u u I o
I. I 0 1|].
I ..I ...110
I ' I
_ _. .
I I I. . -

 

_ .
...-I I If I l .
.I I- {..I III-It I.I.Il
.. I ._.
D. ... . . g . o
T+19+IIOO I

 

.I. b I.I... 179i IIIOIIOII b t

..I. kI¢I¢j Q l . ’¢.OI

 

 

+‘0‘I‘l' OIIOI..I
’OOO-II‘.
....

 

 

v. pr
‘w-{kOI

 

1 I-
O 4.4—9 r-oc-b-‘u #-

 

 

 

L‘IHI

 

 

 

 

 

 

 

 

b

 

 

Qt.

4I$H

_¢_d .
44. {I.I-i
awar..,..uYIovL-I.¢

I.Irv.. o.- OII_.I
I+~m.9.r.r
.. mayo...

. 1t ._IiIOJu
, ..:.w......

. .o

. L

:o L .

.. ..I.

 

 

 

 

 

4—0— >

“-0 v I

+01. I... ..I ,-

MICHIGAN STATE COLLEGE

‘Qofio4-..

Honda.»
.v‘.&-.

 

o
b—c—o—L—Q 6 o O o a ¢-$ ' bf-y-

.n-II—oéno

......

 

*4 o-.

4

o‘-

 

 

 

Y
60

 

 

 

 

 

Y
‘4. . I .-I-J» o—o—I-OII

IIIII

 

 

 

 

 

..I I too...
TIN. III..I.IA II.

4 I

_. 3». ..I-41‘
I’II ...H Or.l

o. u¢amo

I ._...n

A+dno

I-.-I.....a. ..II

o _.00 a

.
..II-OI.O‘D,I
0" 0.... .9
h» Ill. MA I..
... _.o I .
.0 I ._¢
# I0

.0. 4+

i... o.

0.. _.

.‘ _.C
.p ...

$117K

w_

4 14
._IItlrIOI...
0.4.471;

.0.. o.
QOIOIIIOI'I

00. I...

E 0

.m.

M.

 

 

 

 

 

 

4 -4—0 ¢$-o-v4 +6-

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

QI-Jvano; 0..
{5... .I. IOICIOQ .0.,
..IQIO'OIIII fi.4_0 000
'f.. v..-
..I. «o... ..
-._._..+ ..vcno;
IaOI-T.
~+oouIVwo+A
...+~«~+«
-MH-_.Ȣ -
....eo.
III. Id. 44
.-.+Hr.
II... 00
...HH.F
09....4
I a .I.;
1‘

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

f
t.

 

 

‘ u
'QOD'WOo-IIOOQOL
. _

Q
J

 

 

      

'-
11*.

I

 

 

 

    

L

 

 

 

9-0- I 1

 

--a

 

         

      

- _._:

:

1”é—“f'§_

 

>0...
._.-

-O<--

 

.—4«--; ”-9

 

‘

DEPAR TM ENT 0F HATHEMATIC.

 

 

 

 

 

     

 

 

 

 

 

 

 

 

1'
- a—u oar—f v—c—y

t—I—I-o v-1}-‘-‘ o—Q—o .—

--. -‘-<v—u—4p—AI-:.— — -

 

 

 

                   

 

 

 

 

v

A

Y

 

 

 

 

 

 

L

 

 

 

 

 

 

 

 

._..- := 4w” *4

I

 

T
L.

H

 

' ,

 

1
war

1L

 

 

 

 

 

 

 

 

 

 

 

 

 

+4

 

 

 

 

 

 

LLL

 

 

‘ ._If‘ 'O—‘k‘
¢ 4} 4h

0.1—. H}-

v

 

v

 

   

 

 

 

 

 

-..-
A ...‘
A;

Yl
Ié—B-u

4 0.0...
H
OO—IO

 

 

 

AAA.

9‘

  
 

 

 

 

“0.1..“

A

 

A
- '

 

 

 

 

 

 

 

 

 

 

 

       

 

 

 

 

4
H

 

 

1’
U

 

 

I D

 

 

 

+4H>4>

1
to d HAN

;

 

 

 

 

H
«I
i.

v' 1H} 0.4 Ik-é-o-v—o 1} o- r -.-—I -I

 

...—.-:.
§
L
I |
r7

 

 

 

Y

 

 

4}" I

O.

+o—ov4

1P :1-

 

 

 

 

Yr
.0..
H.

--.ap-.:

*1 +1? '-r$-

 

 

0'11 +1—n4'

_I4L4.b o .1

 

 

 

 

 

 

 

 

 

 

rh—o
D

 

 

 

 

r. .1}". 9 v' co—HTI—
ta V1HO+ boo—n+0!-

 

 

 

 

 

L

 

 

3st“
‘

AL

5

T‘f"

0'1? 91}. A

b 0

 

- _ o- 1 ...—To—

 

 

 

 

 

 

H ?" b—- 4». § 0
" '0}:
[

 

_n-yb o
'—-—o$
‘U’.

         

 

 

1o.

 

 

 

 

.<.
L.

A A

 

 

 

 

_.

 

 

 

 

f

o >— Hr—Ip—c --..‘o b ' ffi-oai it:

 

A
.11.

no

 

‘P-OO.‘
v

 

Hi

 

 

 

 

 

>>0.x

H-

 

..—‘

 

4.. ._.-01044

 

 

 

 

A

 

 

.9 -O4
v'vv
co-o—I y-o‘oO 00 r-«J,»¢—4.-

I
II

>4ro—oo 0—.o .—
L

L

 

 

 

 

oti
1
l6

 

 

 

 

Y
A

T

 

2,- 4» ov—t
T" o

'-

 

 

 

 

94}
a.

 

 

 

 

 

 

 

r-o-u H Q-t ' v IH >-- 0-0

bro

 

 

 

 

H 00¢
l
4
l

 

 

“pa 4»
->L
—90-v->-

o -oio-f—4b—0—4

-23

I
ww9vo§ouaoy~

f

 

 

Q

._-_.§$ooo—.AA

 

 

l

olfl
. ...
>y.ro9o~.a~'~

o

 

LA
III

 

I

‘04—,-
.v
'.

....4-4

.. I 9—.

¢O-oo

A_.

-I ."I‘. .‘I 1'”' Q. - -_...-

v
‘0‘.
,4
;.-_..,__-
..
«I.

1:.

}.

0.099..-
._.
III;

 

I—ta—o

FHIUOO

‘1'1MI-4 '00
I

‘fi'

 

I

 

 

v

 

colt-I
0"...uucof

0"

<

 

 

 

 

'4».§-ooc.

-

.._P.
bow-OJ

.>

 

 

 

i1'.
.1.
'7

”w~.**-:-.-.c-I I

.—_.—,—.I—Ip‘-.uo'-~oo
-—o I-I4p —< 1

 

._.

 

J2.

 

O-O .tri

o.+
v

—-—

c
0

1111.1

.¢ if,
‘0
C

‘30.".

 

._.—p n

. 99-: 9 +

\

.-

T

$oo+-co=-.-'

 

 

I‘ll—..-

 

 

 

 

 

I-H'O—r
0| .

t

 

. --+—
I 0'.
- .

 

.._..-

.f‘..
1117

 

—rrq_ .4.;p‘o—o-o-o--p

Y—

 

1h

Lt‘h
..o“

I

WW

 

 

 

 

 

-0
Ln.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

      

 

 

 

 

 

      

   

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

DEPARTMENT OF MATHEMATIC.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

MICHIGAN STATE COLLEGE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 
 

 

 

 

 

 

 

 

 

 

 

 

 

 
 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 
 

 

 

 

 

 

 

‘1 14.141 ‘ 11‘ ‘ 4 4
He 4 c. + «..I. 9. 1
a u. .. 0.... .o o
I .9. . c. “.u u. .1 .—O;--
_ c . . .I.. d I40... 0a 1 ..
1 . und+oololl.d..vll ‘IOIOI.[.OIIOQ
. u to. v .0 AVOOOIOJ .
o u o ou.v.vo_.IIt.cn.-IoA. c I IlooLIOI.
Yf I1|nIJOI+ol¢1Ao ll...I l1ul0lc—.
I. . t:.u¢n§o I. I ....
O . YOIIOIO 05r.m'5- ijIIA. Alb-.411
v0¢1..¢.. Q1 . -I _L _. 9Y0. . ...‘fi. u..A. .. ....
I.Av._Io.¢.l._I. . . . 1 _ v . . .A .... A1... 9. -
I..A¢l_..9.1oo . - o o ....o o I. _ .4. ..A- .fl .. ..
v 1r. . u .. oi... 00 I... ..luu.n, .. .qufl.._»_... . .
v rtoo u... .o n 11.0.- o , I. no. .vt ..o n.- .. .. Y- . ..-
TA “.1- ._. c- In ..IOI.. .. ...” J. _ .... . a. ...»..UL. . .. ...... . a.
0FIJ - a I 0 .u U . u. .c g a. ....1 ._ _ g _
1 0.16 n u . . o - ._.. .. 0.. o . . l..¢..<.v AT. If v

.>». o .._ .. .. . .. -. ...- .r _._...L. ..-.-.IAT (7. . .

4W¥4 W404 0.1.IL I. l]! - .I.-I Ill-I..! .. 'IJOIO..+IIOAIII._1IAIIJ g.
IvAlI' IOIAI.I I I O _ ..I“ .. .... O . ._ . u 0.. . _. _. . ._.
v04n+¢.9I-rL - I I 0,. . I .. ... 4.. .. u. ..w. . 1. fi . .9. A .1

A..onoooa o .I o 0' all .I.. .I_I I'D. 3.! ._.. rIvIr‘Io>.Obn 0 . .0 1
71.30.6000 .1! A | o- . o A 1A.... .0 so o_+o f o. .0 0 _o _ 1. u n ”A
vva 4,-04 I III. II... n . ..I rv .v..ov... r.lb A. .o v. ._. o .._ ... o r..

YAYvIOIIviIO ..I _ , p 0.0 . I u ..I ._. -.:..- . _....fi . u . .. .. v v _ _ e+er
.4 I voI I . v I o uow. Eu ...... o 11 ..IYO l 1., A+r¢0 o. ,. r _
. . A _. _ ___
vArIOIOfl I.. L 0 o. . 1 . .._..A-wI...I $JI0I 1|. A . vO-1rI ...oo ... c .u .A _ ..A ..
.I.. 9 ... I. '.I. .?.v. .I.. . 0 vv‘. .... ... . u . .. .

74 >vhv. vv. . ...: I . u( I . ... +I~v 56¢..II1 . 0‘ .1 4H 4 > n. H.“ HOT?! III+...
IL I 0+! v 0 . o | I . o A _v ._.6 9 o o u a 11 o o b if‘ 0 a A b v .1‘ . .u ._. _. O .
v ...or ._.. p. - cc .0 _._c ....w A v. VIXVLLIAIIr 41 41.o¢¢...o «a
Y.A1_wo ......o _ ..m .- .... )0”....V+F_.o. 1.; H .9. w.:vv.H.NIHAvob.IH.9- L..._.L<..HI_HWI..vQI“.
W I o .... o . . u 3 ¢ QHC$ a. 1 _ . O.

4 131...... ..oIv_ ..I”...6 no v- 0...... rov.._.IA A+v owA J.~v¢o¢4 v . T1...+ .7! or
lvofio.oa - o. . 1 u. 4 0.0. 0.. H‘14l..L II. a 9600 LOOOI‘ .1.._I. 1100'.

III! col.IIb .I.- g i tr .9 to o .0 We" . 1 0A». YI+ATIQOIOI Is ..fl..I_l.\..v...I‘I0._
I.IL..-....-¢.. . .... . _ .. ...... . A .. ‘ a. . I- I 4-44 ..._ .1 r- I..
{65.690§+s&1o..._A r. u‘ ...r. .o; .. OHIAYIO L4 .,._ '4L¢qfi
iii: ... on. I .I.-45.. .IL I: IL 4r 0 o I A .u. TI .v'lll.
v5 0 . . o . o. c _ .. ... . .. A «..I ”..IOIIJ I..]. 4 I.I? 7 79;. AT? ..II I..] .... ._.A .. . ....o ...I..- |I.III._I.II ‘.
Y'POI.1I..OO I... I. II «I o, “O o ou'a 60“. o o. 10.4 0.. 0.0; A 1 TJE lwvu O ..I..-
YOIQIOJ.I Idur. 41f? . n. ....s it. ... I.....Ar_I..,.0rIA J.) O-.. rfivoblI$Y~.P5.-r H u . tollvv.IIo. .dIcI
YO‘Oocooov v. A'c .I« V .. 0a.,v.vll.,.oA a... I.It Phi tf-LIOrv t. K. OI. “I. Iitoué. A
{.1 O o c - o o a c o a o u a v 1 I I. A. r-IQ.A [n 0 v lIfI {Av-I..“..WAT... VI. 0 0 1.. “f a o . I0... .I‘II. . v .A. . . . A
I _ . H . a .c .. wvov....1.aé IOOIAw..IAToI.._I. t c... ... o. a a _s .19.. . . n o
VA . . a... o .00. _A .VH 0v .. . IvarOob1ihr... .I..d. - u a . ... J I.IQ. a .9-“
I r .OOuA*ovo..-r a..v.-. .4....1 “I.I.noIoooLi t ”+4wwwc. . ...-c a 91. o I o-..... A
YA T.IIOIvOoOoO¢oU1 o u§0600.I..IAYH_co .vI Y.v<.o o.o_t. ._.. ... . ...... -. -. .I. .. .. V. . .. . . . .A
_ % It . .4} .0 44vr+ I. I...- OInOrOLHAAI+i ..fo-TLII V Di I . IWoIOI'QI'. .. ... . wilt-..I. .__
VA *1 In...» OLtuAIJWI' ,.I.0I IIIIo ..I'I...+._qo . — ... ... ..1 o ov .6 I.......I III ....-. III.. I IL ..-A ._ A ._.. fl
I . A.‘ .A o. ... .. c a o. . IAA.Y9«.-I....I1!¢.?Ao . . .o o. .. ._. .... ... . I..-.91 .._ . A34. ..M
I WI“? 0 IVA...QJOI1L «Ad. “..I IIOIIOIIA.IW“-vo o no ...... u. to. . . _ I .... 1I.II . 9.. v ..n. 4 _ _. . .IA
1.1.. .‘170‘44 .Avaa. Q‘ .I o .v . OAIxL ..I W ..4lr¢,.‘<> .I.: A. . . + T ... «I I..I.o .. rLI....§ 0A
I L... {I.IIOO... .fi .... ...uoobo. o... 41-141201 .._. o. .h .....o. w. .. .. .. .o .. ..I. .._I o ...“ « ... A
v r ..LI A .6... ...”...Ifin; . .tIOIII. I21 .- . a. 0 o . . o g. ... ._I_. .TI IO. .rIQIIOI: I III [.4.qu . I . _ v o .0 .. u I.. I.. 1.! I.
1 ._“_ Hi r’+,00‘A ,Ihlo ..IO IIOIAI .Y...h.1000 vr.fl-‘.oo +o . u....» . a 1 am. a I .0 A.u‘0~ c .. no. ¢AA
I ..I”. . I.I...II.4 p ... OD ..Vfltu’Av.v.oo. - .v -¢.O.VJ.I.._. 00.0.. .r 9 H . . F r o ‘ n It _ 41qu ... .WAVOAU
v V. v0 0 A .OI’VQIAr A A. A I . n c o ._ a . . . ..A ..I ....l . .I .I. A ,,l. limit! ”I.I .4 I .I... . 1 III. 4 . .1 r... . . . . . . A. d . .. . _ A . a .I I [..I I A ATA
«4 III 1% , . 4 .nfrquOLJuo 'lv . ... .r «JV rh...l 4 4 n _ y I H J . ,A.lru.-,...,I¢..F-I.. 4144.....- ..I. n .I .. I. .I F r n . 4:- |I4L4 0
Iv .o~.v AV;O A..v.¢.¢0.6..rAv.‘11 4 IIfléo H ..I +Avorrhf II...I.II...\._.I.I,IA{..I... .... 1.. IOIL..L_I. {A I..... -.I.- -13.... .. .J 1...”... I.I- .. ..I ._. I ._ . ,b i»
I OY. IbvhtoOI74+.f 1 Tl.¢t.om.. 05.....u09r wfkkooowfl. ._ v ......» 1 .I .9] YII.._OI ...... . . . . . .....¢VAIYLI
...r. I» N ooOI oOIA.A.—r+. 9 0.4 I.I.II .I.-_Avu. '00-. ._..”vru-n¢:nr .I.-LT-.-» I;r_ 0. r: _._o. AIYAI
TA A. o N AYOoov5+10m v. o o v-9.IooooI4IOo 10¢¢vo.o¢oo.qr@¢oooob.o.q ¢.s.o..v...o .. p u A . Irrl
If .IOIOA ..I Irmftfiu QA|§I0IOITA+. A. Oooo.ab.01.o .9100! 3.1. .::..“.OI. ..I ..7 ..OttoI... ...AwI TAYAII
1.....AAAV...VA OAOOLHatr 1.0 ooAIAontovA TAX .10061111I4.¢Ava001o ...to++vooo.bo v.19.._ro.|_ 7.69).; J... ...... .. u 14
..+ voov voOo¢ooct. ..o » A. YA .9. idIoé...Voooo.L..151..oo I 4 I 440. 1 L13... .-1I..?bI-.PI ..... .DIO ......AIIII
... QAQQt so“ rflovoo ATOIIOIO 1114L4-»+ coccuoao1.99.91LTY.T§I.679.¢9.H w r . soIa I....oATAITAI
V4 Awrf9upwvuwoveo; 4 H 44cc; r...A:oo.o......:+.ovoooIoI.,,I¢Ir»IoLIo. I51 . .. _wIm ..--I... . #IrIoJAIIIrA
A H4. I 0xwwr. hp 0 4 I I...» _ «IricIfi_.4 444JPIV twp. 111 V...Ool’0'?...\..1b.-IOI 6.9x .Ly
$00141... , 1-4 ”fliHIbelfl79I600IAAv’ A .H IA. 563%.? I>¢¢44H rII§I4I_v§1e_ 10...44Hoo v9AVAw c. hm....o ..I. t,. .. (.o4AA CA

I 06H. . o . A .. 4A.. fHoo . To ML r IA If. 7. .IAvHfli 0,. fihva .4 14 ¢ # a A #7 4004 4 m. 0 w .4. 0 fr 1 +.“ fifoIo 1 .14” I +4. To. I. III 4 44 +IOI1-._I¢.«.$ . e o -7913 ..I... vIAYAIVA

19...... . . . . .. H ad. L_..Ior.. .. u¢o+. JItAYT¢¢ .Hfi.~ .+o_.HI¢_1IAIIT19-o.¢._¢ ....01417 A. {4.1.4Hoc . 1A. . .....or o.a:¢..¢. ALI
v .1 .WAYAQ..._~.¢0-¢.'¢.§¢1 » F 4 NI . cow I I» ATTH: .OIATovéIAYO .9 To +r¢L or... .ooAIrAfio.o.H4.1.+0IbI.LA4 .1I4 .6106. .. VT. ... . o .. 00¢. .I.-..IvfolII A 7AA
’04P .0....u1I. QI- _AY. vOHIVOI:IrIQIOIo_o .. I dIv fiQtAJ 4 A ...1. AATA 6I_¢I1IO_.I0IA_ 4LAIYAIOIA QIvJ j.II40 4 a 4 A . .LI. . 0 .:4 n . col £i’9044fi
v0.4... ...... .A.v.~oom.ow A A+oochlY¢A1otoow 1o...» 4 0Lr A.I QIoIcAIvv644< H ++iococncooow.v. .AT 99 o .v M ....o,._v_o .94... ...A II
i§0Q¢HItI¢.. 4% TP... ....XIQdIfoIo.o.,.A .AvouHAo .6... ..4 . f...ovc.o.ooa. .W AAv.vo o v ..p. ... A...“ .AA IL

I in:+.IoI —II~4II » 4 401 k.¢IQr.b _vr 1 o . o o ”A o . o YAYJ. 1 vATAv.r¢.Arb_.A A....I.ATAVAVI ....IAIVIolIcl .I.,AA .. o y. A Aw oo# o X . . A. u o . .... A r. HNr

wwwli. . 1N 11.-...Hfl ..Im: ... .....I.- .5 a T...Htf. .... .... . . M t ... .. w;

44 4 44 4441 .....4IW H . ~ «.4 .. III .4. -. N>H LL I.....«I ..4I.I.II4_ . 4 Il._uI...l.1. I,IIl_.I.nIlI w.
17’4I1I6IOQO .I . .A ‘4 . a . ...... A ., w.I..T1PL¢uO...IIIOIIATAw .AT k --A...IA H1A. . _ ... o v . o _ . ... . ., .o _ L A
17.010.98-06. r A I.. IN» o.r0999+ vo¢v.‘Aouo. .::.I» 91.... -4 A1411. 6+4.44..1. ._«fieo o . 9... _ c c 1 . o _ ... 1.111 A1
T 0.9T1'001IUJ - ,. ... 9 > coorOIQOLY.‘QvOoana- uO-.OOIAII ..OQOrOOY 17441...|tv.o._iIOad. . .1 1.» ovochoo +9.... I. . VIOI- . .p
Yb..oro.1ov- r 50-.rr . 7. 6 vlo i7¢iAv1.oovo¢4..tmeo.ooo o Ooofor 0|... 7......InuInI. +I c1..o.f— . .._-.99.: .0.4o»..vr. . o o ...... .9...
IAI.0611ObO.to| co. 0.. § ... OQLI .IchvIO.fOOA J‘M1o.vo¢ 0 0.6.”96090I'IJT.Ot+udno..0..ITolA.v . Id...¢a.nocco o .... . . . .._ .F n. to... o... A VA:
YAVA~¢.r¢1v.o¢ao a I o acorcdu 9' oo.* ....ooryoAYIvIGPIOI-UOHIOQ ... 444 I.I.IO...0- w+r19rooo+hd .. ($1 ._....I. I. u. . .OAYAIYOA
IYLWOIO’OIOOOO I o. I IoI.I.6I.OI.n.._, 1 Av r 41 .._t. toco.¢00t 0.70.004 4 I 01.00.01.0ATIHJ .0t.. 0.. _1f ..r. ."I' .... 9‘ _ _ ItIl
I§+OI0I1IrOOIO 9.0. 901 0......v ‘30.A . oootoA¢o9AYOO.1vOd4|Od 4‘HflflOIn 04¢...6004 c it 00¢ «11 o v n...0.0...I. ...v .d n

444?... {.1491 . T.«-, ., .... C a. . oHuLI > .v 544+ ..9 v.94 t .’ om... HA..1 rot; - .o vAV 9 $icv.‘_._cfiam a
I..-{o to... . .I-IILI.&IY..I0. .. 4 I 4|» 4 F .4r r4 4 4 w v Av ill. . IIIFJIWWL
.. .Irl -.tIvI.h.Oo oo...o...¢; A. o .o o. ...qu 444 omhfih+34 . . A... ....éao . .. .. A
Ivv 9&9... 1..ooo Own-0.1.00: A o .4... .0 9.4va 9.. r. o..?’.... . t. .A .74- «I II-I.( I . ...
I.Itkooao 0 yo-* .. .9. o o 1. 9.... A0 .. . _. ...._ — .uAvV
IIYA‘ffiOIQOVHO co.- »1 <...c§ .0.-..l o .... ._ .... IOAvIAv
VAT ver 0.. ._...oro ._. ....oo;o . . ~... o o ._ ¢OOOA AA A
IIvAvOfHIOAI1 IOOoor in..I”I1.f..r0I.4It'6 4 _ _n . ._ ... . VoAAAAVY
IYAv.oo.o.co ...... .011voo.+.OI.Ot..|l av. .I.-.. 7+0. .4 c 9,.v4 c.
ILvALb6IrOITA .FIAOOIOIO ”onpyufi uh..- .o o* 1 *1.” yo. ._r. a. :fi AI
0 c on .. C O. o .A. . . . I I..
.:.:... .H ..:. .. .. .. _ r I H I. r

b M “00:01 0.4+ HMIYTOMMIO v“ .1 OOH“ A 44.14 “fl Y. OvHW ”NH
7. VHHAH .......o c... rovWVLOO m AVIAL LIT...LIA.[I A... . . . ..OvIIIOIHIfiYWMI

¥r v. 00... .1 o 1 O1HI.‘O. ¢ItI¢IIoI ... . .:.... .._ t o. .1 r .OOOF,OOa1AYQ.Q_O.Av

#rfi 00...... o o 0 ¢. _or. ..o._ .. v .. ...-... _ _ o o .du.l.wufi1+rvo vA

IOIOA ..I. O a O J.&~W Q . 0F c .H_' . .._. o . 0 kg 6 101+0_ rflWQO AAII
0W t00r0+1 r 44 . et....e" Ho .0.... O 3.". no .. a .._O. 0.“.1. 1.90 do bogfi AAA
ILAV¢¢ 999.00.; ., ...r L+A wt. ......o.__ «HH.oooow. AA-.. >L.... ... c. U.” +fi~refioos ».oouA YATAI
+ AAYArvaO .v.4r... ..4-OI IIIfiIoloé A YwWYA O‘HQAAvAfiIAYOw..¢iL....o coo1ooooo - to... . 5A a I....l. fl... 0 1. A. . .. o1orr> v'tofi. xrl
YAvAvAY v9 Fr..VLUIV.AI..AwI..T+IHHAI‘IIII‘II. 00$¢ Ya o A“ ._.. Am ..vow 1 '0 40+ OLA.-. ..I.. FA. «v...1.. firquJOOQ n...¢qu
MA? A ..I F H L .I. If. _ p _ - I ..I _ . .. . . 1
4 4. . HR. wh44lu4 44 4.1. 4 1 4 ...4 ._ _.< 111111.14. . 4H, 0 0 {I'll o I V. v ...Iifl.o.+ ?O*t‘.§.w‘c II
4 0.01 .14.. ._ . ..H TAQoA. A_ .A 1.... Y .1 .<-9-0I0‘L. ._IooIHooAO..1.vloolJ . o o . e c .-I. uJotktvauJJ].
TIVAVALIoa -cvooo.flc. A . ...... a. . A; .( 9L1 v o 1¢I0I¢ll 770+ 00... #91qu o. .. ... .. . o ..oo. . r. o..4..4 4 .o A
{+14 0 .._vc . _‘a . .r,-.. u . o v“. 0 I 44101.YdAI.AT.H.¢a1_o... ..._.. .o+ .. .A .o . . n u . ...... .oo+»HA YAI
TAYAvoooo.a .. o..._o,.. . .o v o. $..Y?A > IIaoooJo4AI§LIo_.r.A_... ...... A...._. . . A I. .U v oo. - a. ov. ATAIAYA
7. +0..... ....o..v_— ... a or . 3.1.: r...4LrAIY.¢oo.+.. .__...o ..toaoouoo» .A.. v .7 I. .o v u -93 . I u. ....o.... o. 1. All!
>?pu0i. caaufcoIHOIIqo . «4 to.» cm A. . ...-co... No.0oro/. out TIA. 1| OA. ..oLI ._. . ... . . . .- .Ba. 0.. o 9 1.? AVAIX
IIrn:61.§-oo 1.1.o._1l.n¢¢0 .:.:. .. .v... >9... I fr A _oOoJI. 0.A 0.1411001 q”... in 4 . .. .._ .. . . ._.... to. ]«.0I9I A¢.~_v.11A o
IIWAva'OOOL10c0_ -*n. no... .“ 00 AT ILVLL .It +4001 Avthoo. 0.... _ fic... » _. .._ 0 .IA ._A.‘. _. _ .IIH ,. __0 1. u.._oA .1”. O Ah
. I| . . > . ..I A at. . . A . r . I.Ill. . . . .
..I. .KuvIl.JI.....1 N .. 11.05.." 1L: ... w: .....4. ..I . A H ... . .I.”. ., . _ .. . . ...... I; ...-
I 4h«u*«.o 4099H~+$s.~.v.+w0$-9¢ LV+L 0? LI.HJ4I..AI}11I~IQ.u_* 04‘..4.«.1.o N. .._-F .Q.III.F4 ...I 914 4.... .o . .*.w._.*m mu IVT
4 0.0...r90 ...AI¢.-.t-.1roono,‘ .0. to... .0. - co...noO!Ioa .I-. I.. . ._ . .. _ . w. . o.o1|o . b o _r..vr1 o
u... _. a. QOJo..Q_o.v11 ....- _voc..oA . ...-1, '00. 99.06.0.o_ A w. ... _ _ __._. .06... ...-o .v -.I1IA+._. 1
0.004046... - unnognv1-.oov {I-I'IO+IA I?Otoodn.¢k.AYAIIv.ocvvo~o0..coc.. vn.- .r-.m. _- IfiIIIII‘II-llo v.0 i..1Io..QIv.o._lfi
rv 07...}; «c. 0. 4 .1:.. cu b!+.ooofi.. OoIOobl1IVIOLr..- .441. ...-1.... r. ,o-v'..0v- ' «q. .601 .+« u ¢<e¢two ... 1H
0 60. I... 1.0 na _ 01-. o fit?oo.9..oI_040 IOOOOIII4I1IV...A.1.-oco 1 o”.. ...-._. 0.01 on. o ,. I 0a.... ' OI .e .1 ..c fl .._
.0.. to ... c00o-.v.¥.¢t+r.oIr.AIHIAIA v.1001844A ¢.o0o¢ooo Hu1.09v1.0 I00... .01.! 1* ....4 IMVII. c .._ .r..-‘ 6.4.011 .0 oH...w_. To *
.¢ooA.oo._o. 4 .0 ..‘Iwéddco‘orHvov ... . ... .... ... t ...1-1. . 70.. o . Anoa A Y
[I v Iv A .9... ..I. .. . c ..rH...... A. ......o... ...u... .._”.«I.I.wII-.ooIq_,.o .... LI...1LI.IAT. 1+
094 _A 3...»: Ifj .1&§.OQO-1hlo.fno.c+v.o.rl . r «p a 00.10.“. »>n
T .A 4 44.- 1,4 T1§70I¢f¢+ «.91... .¢d.1 . a o a. 1 . A .u o. _ n... T..II«T»04.< ..Q «QI..OIoA .IT..rIOA AXII
1.94.1... .1 o A..fi.ooa.+.¢o H10... . ._«r . v.08 4..-«..I I+I ..I. ..I 1‘...oo..o.q¢.+1+.filll
V ».4 4 Q. . o4+1...ocld¢l. .1 .. .._ +7 0.0 . 1.10 OOVI? IIO‘IOoIoIdI o‘clh I “III
01 .10; rt . o A.. OOOIOIHI .n,. on. A. 10 0.; _c .- QQOAL-IO. IOI‘CIV.II¢I p‘IfPVLCA III
’1 tI...‘ 0. ...1-0. 0‘04) ..I. .1. ......Hi .v‘oo .._1'IIOO‘OYA. l0.II...._r¢6 OIOI‘ 0.; 111
1.04101 0 . O b A .._ .%.0o0‘00r. :6 A1... .- v.9 06.0..-.IQI I.kc.0.....O-I-9.'OIOIOOIO IL -.A .11
WW LN»: o t A .o I. a ..I I..I7.IA -_-,_ 10 I-.' O . W.r.¢6.v¢ 9 oovocuouofoQOIA‘AwAIfi
H ._ I_ . ._ r. . _ . I
NH mnww K Ha ” .0 o o. . p. ..w _ _OQIqufioA __.. . 4..... . v o- o...I¢O.IHmA_._.o .o ow.«o.o .0 In
h NuHH MN! .hcw NIIHHM - .. b .» m A». n a» .rF . 05%.“; 0| >0§~ Hum» u an no:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

                     

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

DEPARTM ENT OF M ATH EMATIC‘

 

    

 

    

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

MICHIGAN STATE COLLEGE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 
 

 

 

 

 

 

 

  

 

 

 

 

 

    

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

III I _ _ 1 4‘1
r+4 up» 0. . . n I no. a ..o. - I. H
1c.< ..I . o a c. .0. I o . .F-I I. I I 11 III I."
_ha a I .r _ 0 . I 0 o I I II... I.I. (. I. 1 I I
II A..04¢4 ..I III I .o c u .I.-_._ v- 0 I. ..II
...o. _. c _ o n . a._.. .I.. a .._.l . II II 9
v+l.QIQ.II I.. I I Ir {I.I I Q .(IIF.OIA_...I.Q. .I.-I «I .. . I. I v
Q+O.aI‘I.IIA.II I __ .4 . . .
by I o o q o ”w“ o . - .I.v.4I_. 09.. o. ..
“4 .0..o7 - . . o I. -.II-.. (.:....(70 lo _ o. v. I
..I . I. I _OII.I.I IO va 0. . & I+I _ _ co .
wt. OI.I-.I - I 0 II. I l I III ._ . .
..IYIIIIIQI . I... l I- I I I“. I I I r. ..IJII._I«.IHI.1III.r4IoI-w» .I.- .u . ....I ._ I. ... o v
I-I.II '0‘ I . r O I I I I .I ___.....I . . . ..I .I. A. .. b I. I. I . . . I. I I.. I. .._ I I I _
WIFOIIIVII..OI_I O In I I I III I AIII [H _IHI INHII.I¢.._Ir‘Heo& aw .-.v I o I WI.’VIIOOOOI&I.. .I.... .._ m * I I.. Y .I.”:
onov.o. .I. ..m.. 0 F v 1 I o .I.... ocwo. ...” o 9.”. ”HI. I 3))9vo..-YI .9 ..I I _II I.IJI .I... II.I
Q’O-n' I 0 I .I.. I .0000 O o. I. J... .c n I o. _ . a .0 I pl 0' OJ. 11 I I.I III III I IIIQI .
”IIHIHIJvaI' ..l.IO_I VI ... . II I - [III I.. y... . u¢4a0 .I-_. y...__ g-_ a” My... .HIIHOI I on A V I .I.| 6.0. I It I . .
I-II ..II I II II I I. .. II. . . I It 4. I I .IJIII I. .I I If .
v .I....“ .u __ I p» u o I ~ ”-1 v. I“! uwlwllr HI4-I ”I rodiuww. r .«.v o .- ... ... I c o . I 0'4 I II .I.. u . It I.. 4+Io
v. _ IL- +II| _ In... I I. u. I 11. .I. II. I fiI III-I {I . ”I .I.. I. ..I I.va.l (I ... In. «.0 I . c. __ .. f I .I I u o A 9 I.
_ _ . . 4 I I I I . II .. I. II . . _
I.I...«HII54_.1.II o ....A I ..II ‘ I .. 70 fl I .I....“ ..I I.. I I I.. JIIIIIIH ...rl'nllnuwulilIlIh. ”I.I-I.IFII. I .9..- +.|v I.IIII.I|I.I._.II._II..I. fl... _ ..I
VJIfII._ o ”I... .IIIJ a II- 4 III .11 I II II "I.I-4.9L? lei. J II-.. o I .u . ¢06 k *0 ... a _o u“ . o . H . ._. u “I . v _ u . I. Jill-I Ill. ftp... . I.I . T.-
IHIa‘afo ...... HIH_ . .o L._a 4 ._. s... .mI.IIo-. ..II. ... ..u. ION-...... .14:- 040 .I. _II iii I.....I.........I. I.. ......I .I. .I. I.
1. ..II] :O II II. I ..II. | III_I—I.l...I.II III I. IIIIVI_I+‘I O4I+II. Lird.’ IOIQ‘..C‘. 04.0”...”1: . I .c. .1 ..II”... I
:19}.IH.91£HUHII¢I .I I... 0.. I v IQI.._..I.-.o.v I..-«o. 4..-woman. 4a.: .... Io. v.0 v0 0 who. ..I III... .-
I'IF‘HI...‘ h. 9.4% OI I..“.I_;I0..rfi 1.117 . OOOIII «aIQI ...o __.09 00v. 900. 00.. cu I .U I I..... I..-1 .+ II IIIlIIa
+.. I . . 4+. .. OAIOI ELIII IQ‘I.D... ._OII LrIoII To; I.IcIUIouao .I.. ....o ...I I.II._I .I.“ ..I I
...I. .0. .v0._ .o ;~t¢oev fiIQ ¢_%-c .o.fi.-._n ..Iocaoo IF Irl III .rku-ILFI" .1I I.
4 CI IL IFoOwwOH HAOOYbMIvH I.» .09. .......o. no. _I -¢d_w.<_. I..-..V...
«A . _ I I.. 4. rILII I ._ I.. II. . I
IIHIII I.I.vIo _II I LII-I.I: «b 1...; It... I .HflI q I. I I _ .1— ..I _.IQu _ I. 51.0. u+ 1 o _
.0 .r ..k 1 .IOI O .L .I.. .fi.l_ ..II
I- ..II?!” . I I... .I..-- . Inn: . _ ......flml IT... I.I.II.-- I.
L QIOIU‘IF. IIOLTII f.._'.§ _ 'I u _ WIN-I410 _- 0 I H I.I. .I..“III» O . _ .Iuh IH . L.\ I I _. a 0 L .
v. I? b 0. 0.0.IQQIQ‘IAZ1OIIIIOI . . ...1 l . I I.
I _ _ A- I o . L. I .... I I I
b4...lflfIlOI.I.-IIILOIIP‘IOIOIII1I.THIO+ 111....101011.” _QIJIIIIDI: _. L__l C+I _II "...' 1'._ IQIOII
I. F 1| I II III I I_ .m _ 4 . ..OOIQII I..... r . I -. C I I._.-
.. .. ....III - 1 .. .. . .4... . .I.... ._ I.. .I-...II . _ .. .I- ._I .... _
fl 0 9 III. I II II IDIOIO.D.I4 O I. 0 9.1.0.11 .. O I b ... Q 4 _. . . ._ . I I .
.7 I9 OIOIIIII I I IL. . F4 0. .r I Yo I . . If I II II. .I o I II. d. I .I.
h .. _. . _. __ .
w. +4ovb o$Io .. . .. .... _. _ u
..w“ ...IQOOIA .O-l vl ..I. _._._r.l
. I I. I. II . .I. . . I. .-I...I I III-I
..r;. .I.. _ .. .9... ._ .II
...L.+quo Io»; . _ I _. I.“ I -I. I..
kfl*fiI_w¢_10IOI‘IO _ . ~I L_I ..H ..r-
.. +:I.I...IVII l...|0|9 I. I II I . To _ I It I
_ filial}... I..- 74.-...- . .. .
. : f 1 tau. 1.. 4 - .I.-“:11 II III...
A. d. — . w .r H m ”Hr ..IOWI h. o...+. b | 2 .I_.c
_ ‘I, a ._ . _ m _ _ . . + .7» o.._ b .. .. h H 4 F I _I .. . ....II -.
.._INIP “.4 .H.,I4I. III_4I_I.__ _II..- .I. ..II. Q... I 9 ...r._ o. 4.. ”.0 u I.I « .I _ I.- h I.I
4I*> 1....‘a. .I- . . . .._ I 0.5m P th I . h F 0.1 .0? I.I.LIIJIOIII I . _-
3 .:.:.... ... .:...;i. . .I.4-L..-_.-.J TIM £114.93? . r..I_ _.
”l1. IOIAAHfiIIJI I. I I II. 6|. .. I. r .I.-4 HIH.I0+ r r“ ’Oor. “VII __I I.H|| III..II<IIIu_’.I _ .-
a. _ .....w ... .....o._.*_..; o9.+I.INI-.I+aLI+.voI. T -.IIII.. r . III . . .. ... . I.II .
YoeIIIIO. I.. .r . . Ir __ 4 T to... I fiI, .IOIQIIL . wiobItII oIOIIm . _ . ...
.0. n _b ._ _g I. It I!I.T..II..IOI4I‘II 9000-...at4¢fir v6 ObouLlrtlflo .I.... .._ IQVIOVIIoae...-IH1#I a H .. . .IQI. .
I. u ‘0 can. : . .+..TH9IL..OLII¢ VII IIAJ flfr...’. «I . I. .I.. ..r 0 o. 1.0 ..Q. I.
.1 _ p. It‘llqr .fi fi .1 I IIHAr T. #74. .|..+..IA III. _L .I ._ “.1 .I.. . A
VIIAI p «.IA. «H. I 4 fl . I I :_.... ..+.....I.. I? V . ...
I . . I. 1 J. :. . .-I»- .I.. (11
4 Hr niov I. o... 0TH... H 911:]. IHHHII-vn P r» H
I4 4QI.IOIII¢ .ItéL air» m __ «..IIIfI ..I“. _fl __HIH
IIIIILIII .4o.u..fi . ......oo.. _. L”... on
IIIJ 1.0. .I.—... ._ .«UII-unorbl . .+%y.m=”c « .1 I
YATY . vvo_¢|trO|-.OIIIIIOIA ..IOA.’.vO VIII... 0 : x . 7.4+Ifi
Iffi... o...¢+ Ifipvcu?t4 “uuunaoo *9“ cc. Iv... . QIWfiwuawH+ao¢1
II +; # ++ITQ I... IIT'A . I.. I ..I N u HAv 1.1. M1. . .6 _II I .. r. .. .. ”fir
II I....oleH I 10» 9.9.. . _ vac. v.0.) I.IIoII *IITI. I I.I.I ..II“
.5. *_. . o oooob.+A.v Wt . 4IT§A w.bv..t> r}
-I-¢I.IIII+1$I. “ H . .-I... ...-.- _. H
II YVOIJVJ _oaO.-_7 . _ H1» HI . II $.53. +.w. . ._J... _r.
IVAvJI’OIQIoLo 0...... Odfl" 4 ILDI... p! I» IIOlIIiablf _ _ . .....cbéLrJII 4
IIVLIIoIoIfi... IIIIIII. ..IIIOIIII 11w u. ._ «w III .I _.“T ,OIIIIIIIII. I
IIIIH.Ioo ...... toHIHI4¢ H H . .a»+HHIH-Ipuro o I I... m. IIIIJ Iooooilx
.::.....171... 4” i H . T. I I.....IIIIII. . . .. .3, III: ..I... x
I I ... . .o .o I oOvoIV. I . . V64 v90!0+IH....0IIIIJ AI . ... .. _. .4.r.w._Ioo.I._. ... # .I....oAchco. .I
I L .a I.IFOCuow _ 1 II o .0 vvwéo o?..r 4.00.09..+._ r L. .4...t.o_9IoI..:I_.IAIcI _I» . .. . .. I .954 1.01.4.6... .1
r HM. r ..IOIII 'IHIOIOOII ILW I_1 llL.6.AO.4~<vO k .I. h o DI..HW .I OI+IIIOWI+iOvIIOIOm.4_ II..I..“.I I. I. . . n YI<.I 4
WI£.V*. ouvwhoo I. v I H HHHI. LAolo.OoowI l'rTIL T. +1 IIHHHHHsa..ALHIHIOIOQoIIHIOII 9.01.00de I... . “IwItIur “ TIJIII‘Ioao“. ..II 1
{>0 1 u 441 vMIHHOI H 1 L5“ IO.O%W%IO IVY. OII.H ..II-oro__ o .9..._If. OIJI ? ..I III urI‘IQ‘ 0L44Lzor.‘ ..III
I {IHAVI.VIOI¢I’.. .0 .I. V 6-.. . 1 HH 1 I b h p. IMIMIW .. I... I «..IIWI. ‘I IJIO +IVA O ._ a I .._4 *..-_ IV T11
I .IIIQ I I III I II IF. . I I....I... I I ...
UNIQI‘OII.OIAY QIOIQHOH< I.I.IIOJQ AT“ “HonnNAwHNF N... _. IIIIm.H+ 6.‘Oo#...o_ ...th 0L1 HIIIII F If IHIHTA FI>YII>HI»mI.W». » All
0 .IIOIIAIOAYIOI I III. . ..I... . . a « v._.«4$+4 .. oaoo.;.0I +¢ I........ I s r. I .. II (I.I. y M ...UII. I a
Iv.I+I+IQ+OII.I .TVIOIIfiIO ”IL-fl LHII v6.1 JTIMOO vortL-.. IvI'OII.‘ ovoaoooc.o:...v obI.-vo.'oI. O FIILIIleoIIw L. 4..-r NI ~_t._.JA.._ T 1
IJ¢++vI.o¢Io+.oI+I+ .. ... 0.0.6.. 40.. .o o. ....I. 0.6-11. $6.0... .....ao _. ..I. o o.. .I. . c. . . _ .1“... I111...
_ . «ofoov‘IOIIOIOon... I. 0000.? I I ..a. 9b... ..I-.0.. IOII. IIIw+oIv. ... r . _II
”.4; ......I.I¥...I:I..I.._ La}... ..:.:.»-..figsfin.-n. an 33.3..-. ..... . -... a-.. ... ..::h...-..
4.40+6I¢.. o 60... .9H¢Iovc. ......o.v. IQIIOIIMIQIrIITTIrJIITIIum...:o. o. 8‘ ..n...... 06.1? .. 4.4.9.4 ..I _.IHJ. 1.4.9. «H«_ .r .IIT
I....QLIIQQ o to... .HMI_AtOo » III+H9§LIIIII+IOI r+oo.o;or+. 0+.O I....H_.Iur..u rhfl+ o _. o. In“... I.I” we... __.o...¢no .TLIIOII
. ID o .A II .._ _ I.. . . I In. I.I. I 1 III \ . .
Hm... I . IL»? ..I» $4.. .70... o......»o...fl_T....nII..+o.....Izfi. .... 1. ... I. h I... .H ... ... ..u.nnu uhv cc.
1110. fl«¢?oo ‘00 no. v 0.0.0 LIIIQII Jifl 1'... .I.. F rIIIIJ..FI> I -._» III . L + a... . a A 4 p1
IYfOOIQOJIOOr.IWQ'+O.O.I F‘QIO“. 0. o b. VIII“ H a H H I“ III IIQOAVOW M400... ouIIwUO:H.&I+AI$II1II ... . . ._ w 0‘0; I”!OIOI4’. II III
.. _ .. 4 .I a I . . n . . I .LII
UV Hou‘ovuwu "co—I". oerOtx O‘KLI.’ — 0.70 M I. at cm ATHIHIHAIHIWIOIHHuIHO'FLY..co~n. __ .4oau..._._ _.II. + . H I J-
... .I.. .:-:.. :.__:..i... I.I I I .::.. .. “n“...z..III.I. :_.. , ., . 1- ..
I "H I4 IIAYIOIIIQO‘Io" ......Ic" .... I.. f $.041IIIOIwIO. JOHIQVroa. . .m» ...VIVQIOIIflJYHI I I. ‘Qr. .O.IOIOIO6o _ _.¢ mA+v.f vaI
XY o .f.. uT.L.Io 0.4;”... 9... ....I- .. . .oa.+..r. TI.+6.¢....‘. ...ooTLIH .. . a. .. .1 _ . «hvva.
11_+II. 31;..7. .. a: »I III 1.793....27. .... I III. I-.I.T.T...«.._ r..-nIMHI+.Jn._; _ r I .11
. .. . coo . I 4 .. . . .. .. w. e . ....
...... ... git: :._T n ..-W. . -.:... ..uht. 3...: "33.....4"... . 2.: .I. .._}.I; ....
H» v to ofiovubo I 4. W nflm M} I .— rflnv . _I . r HWIH thud-W“ .IoIaII ....”I. .__ . .Hif #LYII
. I. . .T . it." v 4 0. Y1 Y o. .. o I t .1 . _ 1 .4 .I A. In.-. 9’: .I.. .I. o
11V¢>MA erJMLvYHUO. O+I _ IIIO.4 IILI.cc o o 9"va _Afi “46 09h... ..IIV «..AIHmWHN ”I OQO. It mNuWI‘q .IIOIIIOIII..rI_f _IVIA“. o v.
1 . I . I...’ ‘I ‘ ...
IIIVLZI H. ...uuon ...... 1441- I . .:...... wv+to-o...1.oI Y9... .. "I.I... a .?.I¢o...+4I:.L T..I¢ ..
AYLV.¢v ..Io._oW oHOIHIHIl.I"Iw.4«v v OILIVQIIOIOIII 11 ..IIQIOO tanoovb DI w.— *OOH.$.+. “I.IIHH'“. ac 4 I
_ .l I. 1.1 II I 1 4 I I .I I . _
I40.~HII.o-.Io.1 adaoIfiooo 4M.- .. N I _oH .HHHIIM.1L.LH1¢HIH.H.~ow... vaonafooOof .> o. o. ..___ on #41... ..: .1 L
I.. III¢ o. co. .... .... ... H II t...H H .. ..H .I...- ..Ivfrv+boo* o«J..-....H ._.. I A r
t..<1._.1< I.I. :0... r. .. .._o‘ p . v. 09 o . ...... «III; Itrrraooo...‘ oo.¢.ob..¢7r+ ..fi
OIYHQOIo .o¢.o.s Io... 0.00 H I 1 WHY.) . ar.o..or Yo-~|-..¢;0..v+n+o*oéoa. . ......4 .._... .._. :11
> 1-9%. I OI VIII; Ir- H l. ”VIIQIIHOH . HH4..14?+TvaIOIJIJIIAIotoIIo._.u .#o.rob« TI... ‘1 I
IIIILAVOIIovwoono o-« 090 on~9.Iio.O.+II {I1 I .1“ Io44¢IIII Hip MIMI?“ ......Jfi I.I n ..I—..IIIJIIIIIJIn-{r F _ .IHHIYNILIII
'1an ..I raof.oo_HvIIoI_oIn ..Yw v44+co Y.TrM0HIHI .1. 1 I . 4HH14fiI... .I,fi_voo_onoaoIoOI¢+.n L.” 4 I_ I
Y1 . H I” H chHbI. 0904 o- _c. .00.... . H ._Ifidui M» .I.. "mum. O..«..o. p tr . ...kovmoaiotfi _,c M1 I
I LOIIFo ‘0‘. 9... II ...IoI..oo¢'choc—.'. Ii .0100r1I ...o;.- oHH_.I-t|. ...Ionw I. I .Ioao_fi 0 _o a“, .0.... [I
.I I III ' . I y I I
10?; v<HHrIOINIHINIMHIHq_ IN“. I”H_#r-O Lu. A4; in .IAQfiQI _ o o....5.§.u.600".IHIILIH ”fl". . ... ”HHHHHOW.$V“ o .vO‘tIAIIII
_ . ..I I1. .I 1'... I I. I v .. _ _. . . . 4 . . I.
HW4¢0...0.—. ._..I.‘ ...I,Ivuoxdcwo.+orv. VA. 7” rI".MI4Io.WHI...., .. oHrI.Tmoa..~ .I.. ..H .....ootthu,.-0I2IW. ..IIMOIHIIIIII
... .h --_..-z ..: 4...... I ..:r .Ifl..III+..._.:C. .. I.....
v I. O _ 44 . III. I. III. . _ 1
.i .. ..I 1313...: ”..I; ..:...r:: ...Z___..-..I...-..t iffixu.3.3.2.7.. .-..» n
XIV ..v .v r LrIWIIIIIIIII .5. no I .. I .. .. .. _ ..w. ...“ co... .. .II
. 9.0 . no 9... .
W MILE... 0.. 0.6.0... _kuT . «oo.o_c..w4 ..HIH .T69+.~*oo II+M¥J 1m 0&4..._aoooo + + .o ... .r ILJF
1 I 11441 I our..._u. I‘“.#f no mI'IIIIOIIv ...... Y.vI.HIO”..-oo no.4. ...AQ.0~ ..ao. IOI.IIOJIaI. I. ..I
... 1...-.. . 74.1.1.2; .._ ...-.......-. . .... .. .4-......r 2.1.-.. .. .... w
4 IQIVOII .qooIrhoOor 0. _. IH ‘ u.“ r. IIItdl «In... *9 .906 p c I.. ...-II... 90%... _ v A
r . . IM_+ 0.. .o o. 0.00.9'IIIQI‘ r0. 9. . 4. k _ I _ .. IIIIOIO.I1II
:v?vvvor..o aoro.TY..tln.9o I.. _ . . o a. IoVoo$a . vIoo.boIO o... . . .
fl v. ".0be .Ih +.I_V+Iortogou ”our L.o.ooo.HIw+.IIHIOIIc.I1o «a..._-..0I_H.ookr._.uo +Qa000.0a.uo.¢ouou . ._r vaIIIOWOLvIL AVLJTI
VI... _ .. I 90.6 ..I. o. . . . . _ . o
IIIV v o. v”: _rIOIWIOIQIoIJHIIOaJMI....555?¢IOIO. IHIIJIVIoo.._o_w.rHIOHiIPIHIr.HHIIIHNLWrwoooHcI..~.. .0... ow-QQO 0.-» .4.. In” “a”.. « AH“
n H .1 I a r. .I o _H. .unéfbl ..:qu Hr. .. o o ..:.o .I.... I.....I5wra 0.. ...» . ..I
:11 Vitfi 4 14.»..er IH; » ..~+ 0* .rwwoklv .I. a. I.) I’IOIOIQAIII
It. _or . . I I. . ..vOOJIvOMIOQmoM . -. >I...'Iv _ .I.!«JIIQI _ OI ).0IO¢~I TI
1 Y: .._.wfl.“ ‘I. «on n IHIIIJI 4 0... a I ”WIc. l..Y:.O_ I .04. 447A
?,_ HI OI ..IIQIQ or It IOIOIIT'IrIFVu. o. o co ...Icé I.I .cvo.t.:.~bcuo1 I
cu”.._109v.oco I.Hev1«¢¢ Tic .... 70.9??aol .I....oI._v.I-c I. ...o. ..
..I we 0.1.0.0.I06CII 9. ._..T I4.¢Ioo_o_. .0 I ooOro._-._I. I.I. 5.9.9. .I. . LIV
140’.I.§ VII-...... 9 . ..I ‘ 9 .. I ...-9.. V . 1 41.0.. YAY.
A TIIIIIT .4“. .. oJHIaoI. clon- _... 0!? I.I.rIo a. I. 0090.711
wooHIIW. OruO._ I OO‘FI‘JflIIOoFO'I’OO.’ I....T..II._OOIO ... ._- ._Hv . 1 441.11;
I 3-. 50a.o ....o 1310.6 o. .....tIII. ooh... 0. Ofilflaa W A .01
HHF ">000“... 01 com-._IMO+EI* JO .... fl +0H0+ it” IEJIOII. 1 ..OQOOI v.6- ...Io. < “a d. c
I I .hh» pH > > rt ¥>?. ._.? c P hono.-.—. IIP.ro.I44I¢_.I.qoo......II-uun cww
I ... I. ..IrI.
..I .Isz .. 5-43:»; Ed

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

»‘ fr.“

—.,...-.' .. <

4
, i

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

«1— r‘-.

 

 

 

 

 

 

 

 

 

L

I

it

 

NH

 

 

 

 

 

 

L

 

 

>1

 

 

 

 

 

N'm—‘O...*

 

Yw

 

 

 

 

 

 

 

 

 

b h 0 -4>—

n 1h

 

 

M

 

 

 

  

v

 

‘r

 

1 s—o-o-&....~ o . .

r—.

T

 

W mo—é—o—c —<-— —I

 

  

J

v

  

.

 

 

 

 

"D

j

'0'.

41—0—0

 

b
. —0
J

—"

 

i

 

t4

 

 

 

"-'I
‘;$~¢ooov‘oo-o-ro-

 

 

 

 

Y

 

Y-vo-o—o I-
_I._,__‘

 

 

 

 

IIIIII
v91...

"

 

 

.._¢ a

 

f.

 

 

Qf’O‘v.

0 ._..
-O—0 c-O

 

F

- 4.1—

  

 

b

 

 

 

T

O

 

 

o u—rr—tku-
H=§ o to: t—-
D :—"“1D-‘=#‘. 04-4

 

49.1».
- L0- sub-1

-+r-

 

\

Ar

 

 

VfiF."“‘O
+L9v9¢oao

 

‘ “_II-‘

 

y

DOQOIO'<O‘

 

 

 

 

 

 

 

 

L,

 

I

 

I

6
I
T

l

—p-o o—I rt 1
é—n-o—po

Y

1

1

g.-.
‘1‘?
9 II.-

 

.
-.o—b
.,
‘
.3-‘.

o
’—o— a—b ‘ .b—‘-O

h . . . '.
J— _._IA—JJJ
;&-

V
I
I

   

 

'1? ‘r

05‘ ‘ LQ-Q—o-
I
(

La

 

4.00

{55.

4w-
.1»

“—o;|_-‘- c—oa—L—o

.
-.+-‘1+—- o c .<
‘I
_._
w

 

IIIITIIII

Y'VY

 

_¥
1

 

cl
u‘fooo'.

1.
I
l

 

 

 

 

 

 

 

 

 

 

 

 

 

7
‘- .

 

 

:7

“_+

 

 

 

 

 

3
I

 

 

I—OH 5‘.

 

 

 

-—v -.---.- arc—+1

.L

 

 

'1
L

 

 

 

 

 

 

 

 

 

O.

 

 

I

 

 

 

 

 

 

 

 

‘1
t

 

 

 

 

   

2

 

 

 

 

 

 

 

 

 

 

 

 

i

“fir-1 $.—

 

'C

 

 

 

 

 

.¢%.4Hh Iota—=4 .. o-o I

 

I»;

II

A

 

 

 

 

 

 

 

 

v

 

 

 

 

QQOMQOrI

 

OI

 

 

 

 

 

AI;
VVY

 

 

 

 

 

  

 

 

 

J_‘

 

 

 

't'or-tw o 1-1*

fivvv

 

 

 

 

111
IIII

 

 

 

 

 

00er +;4 i

 

 

 

 

 

 

 

..H:

 

 

 

 

m
WL'I

 

 

 

 

H

 

 

 

 

   

 

 

7 din. :JITJTHJY

 

 

= o ov—‘v-upi-O-

v cw — -v
.>—¢—.—.:

 

 

 

 

 

 

 

 

 

1

 

 

 

 

Y

 

 

 

>$f~b
o—g o ‘+‘1H

 
  

 

 

 

 

 

.OILOOOO A
.I. JQO‘llfllle

009
rt»

ir

 

 

‘—

0

 

 

 

 

 

 

 

 

 

 

iATJY

 

 

 

 

 

 

‘ 1'

 

‘ooongr.

000011-

  

....{1
P.
H.

I

 

 

 

I?

41113:

 

 

 

 

 

 

 

 

>.
4 0-4
15--—4>-:‘

T

 

 

 

 

 

.4

 

 

 

 

01+?

'.-. 0‘

 

 

TI

 

 

J

 

 

 

 

“.1

 

MICHIGAN STATE COLLEGE

 

I

 

-o-+—.-
' I4

v

 

 

r 1—1

O—f'O-O

...-nu

 

v—o v-7« 6 b

v

 

  

cr-.-O<
:0. 0

—o

 

 

 

J

 

 

 

v
A
Y
A

 

  

4: f..- ‘m‘

AA

 

 

.-o'— 9

 

 

$~~IH+U '999
H-babf

 

‘1
”III ; .
41

 

I’il'

 

 

 

 

' 1»-

.1”

 

 

 

 

r,,,,Al

Q.

  

 

'-o.-a¢=

to=—4 4.

v
.¥

vvv
A

 

 

 

 

-.

l

t...
l
-“'.C
O
44

 

 

Ii

IfI AJTI

 

v-r-qv-
‘0‘. D
J.
l
0-

‘r

 

 

 

’

0A6 60!

 

 

 

 

B

 

 

1".

 

 

 

 

 

 

 

Jp t1 k-tr—l

 

 

 

IIIIL

O 4

 

+-4-.4>~¢»I Q. I O

 

l

  

' 4

 

.4H 0

I

~-.—-4>-oo-.o..

 

 

J.
l
cow
..-4
-

 

‘ O O-t-I -<h—1>—a

 

 

 

 

AlAA
>0-

 

 

 

 

I I
1'. ._A

T...

rt..-

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

. ”.4

 

 

0.410}

 

 

- ¢ 0 v—v'fir-o—o-

.._.'..

 

 

 

auto—

 

 

 

 

 

 

44.153...
-:.4éo-.oo

 

 

 

 

 

 

 

 

 

 

 

A

 

r.

I»

 

 

 

"*‘""“ 01

 

 

 

 

I

 

 

 

 

 

 

III’III‘LJY
'4‘ I
. u

H .,

I M
. TIIv
5.. OAYII
9 0L}.
o +...
v
931.:
«r

>
...

W“. A 1
u.” c YrLI
>9 .. LIL
var 0+.O

. -.InnFIIM LNII

141w4Hw

x|.|¢ 444

¢4A¥I .+ 0 .II
+. H .o I I
. >

I 1

fl u 0 VII

$q :- czb-orr-

 

  

 

 

 

 

 

A

LL
.-. 1 «air—4

«no.5...

fr...

 

 

‘hdh .q-_ ... ~_v;v_.,..._‘ LH—O—f—Q

+ ‘fi r-«-1-n>0—O-{

 

_._.“ fill-I..“
l LL71
I TfI I

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

... 5.; o H P.- ---.-‘ ...

 

050010054-
"V‘I‘O‘OOO

 

I

'I

 

 

 

A

 

 

 

 

 

 

I4
Y‘.
fir

 

.14..

 

 

,...,..4.

 

 

 

I '

 

 

 

 

6

0—0 -¢-'-OOO<>~

._

 

 

 

 

 

 

firou+ -.0——

dry—'0'
II '

 

 

 

 

 

 

 

 

 

 

 

 

 

...-I

 

 

V

O‘OG—J—o

 

 

 

h-

'Z;+.++;
:1

sub:
A

 

_._4

 

._p-O—
a. d;&..ooo¢

_.4

 

M I
._...IQI
4 t I
t—Iv Av
{.614
I ..1rl III
I. WL§I YTII
? ‘94
ovOoA ogl
v00
W...” A A
II. . 1r...

 

{-nuuoo’o,

;-,
V'

warn-”v r. _t.- uo-o"
H-o

IT—U. o .obo—Jaw.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

DEPAR'TM ENT OF NATHEMATOCI

 

 

Em;

 

wImHI:

 

 

mImHIo

wImHIo

 

 

 

 

F—T

Pods

oImHIo

 

 

71

OJmMIo

 

:onIlllono _

Moono

 

 

 

 

wImHI:

 

ET:

 

 

wlmdlo

 

”-mauo

 

 

oImH-o

oImHIO

 

 

Stems

 

 

Daily Average

M83

leMflull.

 

 

MICHIGAN STATE C(“IILECE

 

 

mum”; .

 

mamalo

 

 

QImHIo

 

 

 

Leaves

 

 

 

 

 

 

......

 

IIH- -‘\HI‘ME'NT OF MATHEMAIICB

MICHIGAN STATE COLLEGE

.« ”m?” .V
. I ‘_ V

Pods

 

( Hoaiv...

A. B

”I M.

..,I r

I n m

I, I ..

e v. a.

m 1

M

W

I
8
n
a
.u

9.
wwmwmrt..M. snuggled u

 

 

.w; mwmay:

 

mImHIo

 

 

Olmalo

 

 

Macao

 

 

wImHI:

 

alwalo

 

QImHIo

 

 

F. Mango

mIm~IO

oumauo

DEW

 

XXVIII

 

mImfiI:

 

mlmalo

 

0.3-01

 

 

Moose

 

mHmHI#

 

wlmfilo

 

oanIO

 

gonna

 

33

 

IMImHIo

 

oImTo

 

x33

 

2

nomonaan m

 

I
_J

CVIVJ‘RFPTh‘IL-LNT OF MATHEM ATICS

E (-5 F

L "“"l.[

IAN STAT?

MICTHM‘.

>0.-—ol—v- ...-.. .
n

.—

Bods

 

 

 

Daily Average

mum»:
mumauo

 

013-0

 

Leaves

‘Stems

guano

 

wumaunx

wImHIO

 

oanIO

 

Muonu

 

 

wImHI:

 

wlmato

 

OImHIo

 

 

Mano

 

J

OF MATHFMATJCS

NT

[-Li \‘R’TME—

,_...-I__._~L ...- ... .

.......

riff"“~w

MICHIGAN STATE COLLI’T’”?

I

. I ”7.7.... -.w'- v»,.~-,_.._r~fi.—.- I
‘ I

-~ “ac—vfi-v—Vfi why-n» . H.
. .- l I

Daily Aterage

Pads

30

 

j

mumw-:

wlmanq

 

owmauo

 

Stems

25

guano

 

wImHI:

 

Leaves

wumauo

 

OImHIO

 

M83

 

msmflum

 

mammuo

 

oImHIO

 

guano

 

15

pnmdo: mud m

10

 

Ir‘ItNIKIILU

«'I I "‘ILI‘III pr Mix

MHSHHSAN

STATE

COLLEGE

5994‘0’15

-v~0~

Ov‘.
00......
~~6QJOOQ‘O

 

._.—+9.

-v—v—~ O WY-v-

 

 

co

-.....

Y
v<y+ro¢qo

   

DEPARTMENT OF

Q‘noo é-v
*9-5~O‘
00‘ O
._.-.-. 4.

vo-oo—o§o‘

6

 

MATHEMATICI

MIFHICAN CTAT’F (“Pol I Ff?

.__-q——--. —-—r— vW—mo -'— -fi~-u-‘.~~’~v ..

I

 

~p-—.¢—w""T~—’V vwmwflfi m7.mv - “Wm—”rr‘i.
. . . , . . . . I , . . . . . .
I . . . . . . . . . . V
, . . . . . , .

XXXVI

V..v

Mean of the Daily AwerageT for Leaves, Stems. and Pods

 

wImHI:

 

wIwAIG

 

0.9-0

 

20

16

232. E. a

x83

 

wlmatz‘

 

. .V... _
I‘LiIPI .III

wtmato

 

OImHIo

 

x028

, ,
. ., . . I I

v I 0 . . . - ‘ .

W.‘ -..1.‘

I.
O
.._—._. ‘

UL! :\'|ll

 

12 I.I .

 

 

 

 

1d

 

 

 

I

 

rm. .«4
Too T

I v.4 II
4 .ro

y f.
01?

o.

3

AA

 

VTOOOUOO‘w 1
{Our anI-IIOIOI I

J---

 

 

 

::i.'? '

 

 

 

 

 

 

 

 

 

 

QOQO.

1: -1.

 

1

 

 

0L.-.

 

 

 

4-1 a...

 

I

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

040.41:-

 

o—J 1.9 4

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Ty; “I.I
Y. 0.. .1
Iv001 0.1
”I? W*
44
IIII haw
v. c _,
o
YIII _-
ILIY ..
I . ...
1+
r+
Y.
0
Iv v .
.._
O
r. VI VI
rm I v
I rv .
fr 1
II VI 1
IIV+ r
IITO; ”
W k.
I o .
114:1
II a
IT: a.
l
Imnmn
I I 1.
I1.—
IIv .
117144
HI; .
I. V.
TI ¢¢Iél
I 1...
In ‘00
III...
171.640

 

 

 

 

 

 

 

 

 

 

.Jr-

 

 

 

 

MICHIGAN STATE COLLEGE

 

 

I
I #1041
Inc...
T: +0
1H1.
II...
Iv++l_4O
ID D
41.1
IVIVOMO
LVIYOOO
I ‘0'.
r gr.o
1.0+.
II
II
II

 

VVII

-: 1.1‘““+‘

1‘ .,‘.'.-,.;.1.'_V._

I
o 1

—< 00—...

,0-
tb-

 

 

{I

III

 

 

: I

 

I
III III

       

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

    

 

 

 

 

 

 

 

 

 

 

 

T'IF DIFIQI-LRP‘W‘IOI I 90 4

 

    

   

II. IIiILIIVIrl, Iv L

 

 

 

 

 

 

 

O
A
Q

 

II ITI

I I.

I .II .I.

I v I»

700
I I .L
. ‘1‘ ..I; o A 7,41%.
. . IV 9 .9. '1. .
I II i m in
I 7, $1.. I “I _ I
... +09.»»I H % I #0 1 1‘
.. V~.r.cwa N + I «I our
.r 45.15... I: .+ o. .IIV
. ..-.~« 9 wr 4 w 4.1.

.90. > r I v.

1‘ Jr“ w+I 1. o _

a I 4 I _ , . I

a .. 1v... [_I «W I-I 4
V,.. I..? ..., ._ 04 >7. ,. vIoIl

100 I.I. 04 J1

HIoIHIr ,r u“ 6+ .0.. . I..

v F ~.M_H 9 ooh. .

L1b1 169* o o... c (”I.I-cpl...
:_Ir “L *1. n . 1‘4 _ . loll..-
7010... I.. ._ 45,- o I.I-Ir:
will ..fl.) .1 I
I‘lfi1fl4 W’Wml :9 1 0

will 10, . ’01:“ .c .9 I.IT‘
.V ..I i OI va . .v - rut . o
II4.“#A vthulfwli v—‘L ..
ILI1 It] (It A 74.. I‘d o IOI..
110&¢_ TI’O1 A I. _
. V. 1 J

 

 

 

 

 

 

 

 

          
     

 

._ Ian. :1 41

._o,§.4 .411
.I D _.,

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

      

 

 

 

 

 

\‘I,—.
«m 14
d I.I;1I.IV—4A 0.. I 010.4 v ..

 

 

 

      

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

’
_
.
.1 +_
.1 1
__IIV C .I.. ..r:
_ r6 BLI- In! 0. o. libiI -V

      

 

.._.T.

1,..I.

 

 

 

c v n 1.6 I. .0 c
_
vIIIIAI’ i. I. II. o .I..-

IIII1vII...IIhI

  

v

I

.I.

r . I .
..: '- ‘4 3*."

.I..-.~H.
J OIOIV'"
rfv.:..
10.0.19.
.I.. ..IDJWII. I.I

 

 

 

 

.._fr0641 .11. 1.0.0
4
.r... I... IN“: K ..II? «I: I 1:; to 4
. _
0:.1 . HI I ..:
.‘I’vtl k.
Olive Q.-
. .
Y. I.I L. .7 + r... I.I.bl.‘ 3%;
proforUJ .f. I-...IZO.
Ir .brbV 4 4. I»
Ifi IOOIIVOI. .JIOIOI-Ib 79.61.10;
_ v 10.0.».ro r4u¢fou‘
._ Yr). 0" bIAVII .1

IL»

  

 

 

 

 

 

 

 

      

 

 

 

 

    

 

 

 

 

            

 

 

 

 

 

 

 

 

 

 

 

 

~ >
1—0

 

I

 

.II I- ,
r
Lw r 0......
f ....4. .I...
3 4 gun
‘ 1M... 4+1 vY_ItIIILII..IHIIb IIIOIILC I
.rr 1.
luII I..“_. 1:.
h ffi v+ 10.0!
4% IA! II. I.I..I
.. or Ii r- ,V
i I}
V. a 1' .v IOIQIIA
_.l_. ... MT.
1.0.104 UT Lib-L
r) I.I I
HM HI IIIIV _
m IL..— II>P L» p
4 1 1 I .—
. o V V :_.. . u
I¢¥I t: L ._.»... I‘M)»
v; E I .. or? I .1-
5. Ir H. _ +9»... «a
* . +0. _
I .H...
by

 

 

 

_"

 

 

4

 

L'. 1L1: o --
J
_' I

 

 

H. > 1&1 o o—o :_.—I

IHHI - p41~14

 

 

 

+1

 

 

 

-l-1I-4
A

 

  

I

4
“I

 

I

 

 

  

 

lI—A—I 4

a.

 

I

 

4|

 

3 =0‘ 0
iIR-f A 1.-
, .V .

 

 

 

 

  

II:

 

 

 

'o—ov9'1—c |--
:n—oo'c
0

 

 

 

7414 u..'—‘Alv

 

 

 

 

J

 

:3.-
II

 

"I

“-::.-.:...7?
..181
.0. «I,

O

V— :—v1—.—

.4

 

 

 

 

 

   

'flOOOI

 

o-“

 

 

 

rv
.‘I
J.

YI—T

 

0'

 

o-poo

 

(-4

 

t

 

 

 

 

‘0‘.

I

0.01 }»

 

._.—1.—

90¢

.
“In
.4
VI I04
1 I V y
4
v. - .0
I _.Ixr
.IITIL -1
,. . .
III. Fl.
1-

 

 

 

 

..Jo r

 

II

91
'

IT IIIII

 

I

T—

A

  

 

 

f?

.._

\

 

 

H" '-

 

00 >1- Ono—1»

 

   
 

m.“

I..,

  

 

..1'

"......1-1-1n0oo

 

. LII ‘
tut—11

”Ti,

c—--a

 

 

 

I
_L.. ...V.

a...

fill.
+bu‘n-oh

.1

U
C
.

—._-1

.V11

.J.I.
O

3.0-to-o1

J
..+-¢‘J.~Q nr—»

_-. 1n .. ,

L

_.
T 036...?
. .
_

. ._ _ .
I'I‘E

 

 

J

 

 

 

 

 

#4—

IT
)0-0—7 I

 

 

:04.

 

 

$1»

vvv
p0... r1
turn—4H
A.

o“.

 

n—oJ-A—u-v

.41—

’1

-1_--_ ‘5 I

O."

 

 

 

 

 

 

*9 h‘1
v-o—‘-- 4H

 

 

 

 

 

 

 

 

 

 

 

A

 

 

"T _

 

I

 

 

 

 

 

 

XVI

III

 

 

 

 

 

 

t.—
6

  

 

I

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

. , , I..-.._:
to... ...r4JIVOI IIII. .I. .

 

 

 

0104'
..I

 

‘Ioo-aoa.o~—&o1

1."'

 

 

 

fir

I_-u—-.-—

 

;..- 41M”:

 

 

 

 

 

 

 

 

I
..hy

 

 

 

 

y

 

 

1 r-co- 14 'T.-—l

...-_‘IIf-F'

.. .._.
...I
-oov-4

I1

 

 

 

 

 

 

 

=2

 

 

-::- 1 Ivzo'wo— .11-Vb

od- 0 0.1

 

I

1104-1 -b- D-C

 

 

4 u “I;

 

 

 

“11

 

'1
1

 

 

‘1 .‘+f'C‘—I 1

 

 

 

 

 

 

 

 

 

 

 

 

DEPARTM ENT OF MATHEMATIC.

III. III IIIIIII. III ! jl..II

 

   

     

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

 

 

 

 

 

 

 

 

 

 

 

 

  

  

 

 

 

 

 

 

     

 

 

 

  

 

 

 

 

 

  

 

  

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

    

 

  

 

 

 

 

 

 

  

 

 

 

 

 

 
 

 

   
 

 

 

 

 
 
 

 

 

 

 

 

 

    

 

 

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

MICHIGAN STATE COLLEGE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4 1 1 11141 11141 I 144 44444 4 1
4 4 1 411111M1M <11 11 1 .14Ifiq41d 1.1"..HIMH1 “HI 41H H HHH 11 I 4._<4IT<.HHN.I<M. 0 0.10“ O 1...! 0 00.1 :1 ..I. O VOW)”

14. .. ..I... .. .. . .1...I.I... ._ . T. : C+$ .w . I .
lonHHwHHHHfH1I11I_o H 0.00 101-... cc 1*...”1ao faa __1 d a ..V. 114 WWW L1 vnuu .

910. 1 A _ nolilulr 01.1 to ._r_.- ~—.¢0_ .1 .nlt._—04I._I OFIII I.I-tlrL _ .
v.19.xfl19w04V.~I1HM-:DO..I 11.1. 1 ‘- 1f14IV4IHIrI§I§_._. O.“ 1 «yfifi o ....11 9 L11“ .1 P If 9.

Iv. too-1.11."..|1H_LHIV.....I..1101I O 6...10.|011 . .0.,»I1If1 J. bII'rroco 1v. 10fi1I_ *CuVVW 1.- .0 01-0.“.

1111 O 0 1 0 H v f ._1 ._ 1 1 ..r41 Old 1 0.! .1 -.b V. p 1 1 b 01H .11.. t _.101 a 1 1+0 1 V .V I; 1. .I I+.l_ 1.. V. .I 1.911 P. Y. +II N.) O

IIYOODO' .1. t ..r _ _ '1 1 1111. 1. 0+.‘Io.1.YO.O-¥rTOI Cor 0+1 1... 0.4- ..1~r++..IflaO.. +0. 0 ...

IYroo 110.00 ..V ”.101... 1 ...... ..I-.....HIr-«.1HQ._0.rI 0. 01'.v.w9on.91 1- 1 1— Hv$r1a1w 1.91 . 1"?“

1‘... . In ’1 1 . _ . IV I u . ._I.‘ I a V

nxVI......fl.:. .. .nn..:+.....x.. ......F. .I....LLIIIAF. . I.. .... $5....) 13. ..I.

I <191".4u1'O1IOHHHVfi 00.0.0. twcHsriofifim MM '16m91vovafi I. H‘LY: THIHtlw+509~IVWW§+II Th":—
v7 1 ”VH9.“ Jr.:.. 1.. 0011 1119*.1 H .I.. DIP.01 H«% I .Oo1co._v v_ '01» 0... Or
I.» I);IHV. . my. 111 .11 1«+.HOIW.1I§IF0.|+IOIOIIYI: I111 “1.11%. 1010 va-wouon VAL
:44 0. .011I1\0_0a .._...OloloOdIo. ... o1 ._. ~00..oro.0I0.II0.thL. 4.. ."V.+V1Io1: 1... 11110010 . , L ,
.VVWM.H.OV_ "_VrIOIO 91H11_11H1_ +1V01‘fiVQlIOOIT. ’10 oasHo w.1.e|1 r 110‘ b. . .

I ..I ».__-IL r". _ V 11 1T1..- 0 VvTIVIVo 1Q1 1.11 1 1r.Hcto~o .01V.o.. I 017171-11 “..-? . . 117161.4311-

r 4a.; m 1_ I...V...o+04. o. 11119.9 H0 . w4r0.ro.90..luov...00...0.ro.o1.4. Yt.v.. .1+.Vo~_ . . . , . V _ ruf .o.V_.v.00V0.0'.I1

T I. .1: w 4.. L+.wr. . 1 ”.11. ..I ....H».... I .....-FIUUIMYI 41 .1 or “.L1.+-...-....aVV V. . V . V V V V V ..T... ..V... a -.....3 ....a.%...........!.ffo.

. .. V . I .. . .VII 4 .. . I .-II I .-.I I .... . .V .. V. ...l .V-... . f.“ -..- 4 I I . ..II...

Y $1104»... 1V « M+ F4 5 H. 1 44 o 4IIW I I1 I ”I4 11“ I1 I L .I1 ..I. .Ifrt-J . V . . . V V . 1V..1,_o__4..l..4r .oo V.. ..H..IY0.10IQQI10T.Y.II

\|\ I II ..I .I.. O. O. I. III}. . .. .. 411;... _ ._ . I _ .I. .
HmeWALH .14 TL... 9%.. I. 11.0 1 r«. “......II- QQIOHJIH! InfiroIF “I.."..HOOIOIHIH 0 r+rf ..ItnwufibuiolO 4+.I.V 1 .11 II....I.._1 F.... .._P_...IFI...-_I.. . . ._1 VJ- 1.9.0! .10 .09—1. 1. .%~
I I0..- 1 V... _9r0o 0. 1.19.0AO _ 1 11.0 I. ..IIIOIOIO 1a._ I ..I.. I... If IV.0IIInl fri Iovll 10...-.I1I0.00r00 ..I...0h. 4.0. ....V. . ..1 ..II ...P AOIQIOIOIVlwlf '
4I_r|4|F#IO. .1..10 ._....f“ o no 0*. 4 11 H.9FO‘ 0-0.... 5 4VH111 TwruT+H1 A u 1””. I..»IOI .__.__ .1. ..I. .vl Ltlnl_ m HI. _. . .1 . . o .1 V a.
1 4.4 _ u v o 1 v 1 1 1 o 1-1 m.IiI u 0 01.1..0IAI 0.0 .o v . I 1 . _. . . . . .
Irvhtmwomnrh IvothYHOQM1+10HoIhIK. I _II _I'OIIOIvlwwlrIwT-w vHFw‘4041. ww .9 I.....
I1 . ... [ILII h I 0 I? . I I.. V. .
IIIYto». 1V1... IV-WYIV.I1wo. 1 0.01.1 . I. J I... m “4+.“ 11.. _ 1.1. _
”0 .0 c 1 ... v-LQI+fH.0. F . .760.I.10+0+1av _ _ V w m 1 .
XIV AVHAV 0” 1 4 _ LL._IH.1— o ’ o I Cl 4? .0Il... .10 5+4 ..Jl'Jsl ._ . ..I% 9 9 .0 _ 1.. . _
H H$wu 1+0wuoh1¢IIJfiJIOI v.6 IOTIrwulluTVIOi 1. IPI01VL- .1.|H.OI
J - . ...fiL H . I- .::.:z. . II... 9.1....
IIYH+L HHHIHIIH-.. ..1fiI+1 «IL. I H YI Y .1. H1 Hu11117 0. ”I’IHI6._.#I1r.V.
.4} . I 4 u 09. 1» _ 00. . II. I.. If ._I

ITrflwrrIWOIO- AID/OpttOI .. ."LTWIHHHI' I199 V v 100.r+4+1fif.I ILIIOOQIIOIfiINMA.

IIT414I§+flIIH. I.IItJHI14 .. — 10 wQIv 11.510... a 41+ 009 . 1IIHI0..*I0.' .V..

I- flooafl' OIOI'J - I41I +0.10... YA 1.1 1 rd 00010+J+9 *0}II.1|0 .I. 0.0.I.JIII

III 4 .A_. IN]. CHI. 4 V ..I. r . _1 0&14 O fib II I _.V 'I_ .A . v o _. _ v.1 + w I. u a. V... 1 ..II.

Ilfifl411+0“1'._lv_. .I.-LI. V_ r II. II. 1.. QL. afiIOI 7. 0.1.91 4. ._1AO-01_.. .wIH 0.0IAIIVII10I09.£4_.OJ.I00

II FIJI». .H 9.1.? H... I I. r w .__1 *_ . _ .. .. V 1H9"... . _.f .09 ....IT I .I a F C v .V H“ . 1_¢*.H I? _ I I... .10 ..V. .0 o u +1..|

IIr 1:4 ._.IIv.ul'J I ..I“ [II V h .. ..r n Army-JILIIIII ..II.- _I_. Wino-Or II. If _ . I 1 H 4 .1 I.I!

— H 41 _ .1. _i ._“»__L!.I_ “ m .I . L.0v ._.. ~J _. HI H11' 104 VH .I...I1I _++..C.OIVJ..MI'I:JI.IIDI+1V..

Ivo_.1of m*.V1 _L. .0” _rn... V_,)..1 .voJu Li ITuvIolwnIu Idl. I- 1.001.. _+.r104 H. .n :._.

T.‘..0It.b "_o .. III H «”VII I _ V — .0 I b - ‘Ifluolo Yd LLIH«.V.+0 I... 0V 9.. "_1 _+

1.9.1.11 e1 1111 so . + .1 t 1. ‘0I9L.IIIO.-0I0-11JIL.II.II «#1;

191+. 11 .V. .1 1 1 H. v... 10.011.46.0I VI . 11.»...1111IOJIIY

1916.1 1 1 L n 1.. a... 1 1 1.1.0.1.- 110.. ._ 1.

101.1! “1“. .. a. 9R . ... 1 or .1 1 1 .... .1111

ITO. . . .1 . . . .. _.+T0.0-0)V .. 1P1... VI. .__. 1 a o

I; . .. .. .. I- - ......V...

. _ .V a V..... -V....

YT‘H H y. .- .f. ” V_.\.|QIOItn. .1h...fi1l:....l+.o.w-.l

IOIO. ... Ir . alt I.I.IIDIO YDI... ._ '1 1.. J 1.. 1

I 1 1 1 0. = I I9.’ 0.0!...vuII1Jm

IV. . .1 u =1 1 .1111... .11. n. .11

I 1 1... _I 0 H _ 0.1 u'AIrIO v 0 LIL... L I. o

I... 1. 4 . LHOO ‘ ‘1orO; . 00 I

u .. .. . I u. . . _ a 1 1 .

TV. I ~_ 1 » 1+ .. w IT. 1 9.... 4...-411:5 .vwl .. V 0-4 .

U $1 0 I... .1JI4.+I I 1 w 1 Q1 1-1.200 00....1 . V .1 V

Inn“ 1111.mo_r: 0 V 91111 1" I.» J“ .

I . . A .:.: .. um 6.....I.

uHWH 011.1 11. 4. V M ...Hbo-o1 I Low 1.1.1? IVWLIOI$PLI7 304+ «_V ._ L011 _.

F01IIOHHI0r1~1£I2 .._ .1»10011 1—4.o-11..._. v 1101 A.r01rv0+dl-w¢u4f +01. cowofl. .
. I II. . . .. 0 O 01 Y. 11.4. ¢. pInII _ ..

-.-..Tz..._.:...... ...-.I..-.It_:. “.I....” Luna: ......3: ... .:.:.. I Ir. .3... -..

Y to.w.oo.o 111%. L. 1 n 1 .1 V111. ”41111101.“. owfiwo . LIITI4...0._..0I.... I.....I Yr. 1-. _H .uv.

I IvIIv I... T+Oluo ._.. .I. 1 a . _ .. I.. .. f I .II .119 VIOO 0; .I. YOP I‘I .. b.

.-v. .00r+.9110-_1v.. 7.011 .1 1. 11V.o.+ . 101:1an V $.03. I .1. .. 90.00. 9%
I .. HIV .1— +1V _ooIM..Io 0.0 .1101. a ”1 1~11111+$A71._.10u1. ._..1.L.1I1.1r...?v+.o.fi .91, _..oV.._0 .I..
H 4r-1m-11!_ - . o . 0.1“...» ._0-1 111.11% ..fiawvcoao ._.Y*w.fl_..4.0 .HHII ..0. Q L.I.JI W!
H II _ I.- . . .-I 0 «III- . .1. I.I II... I . 4 .. .
Y “.2.“ .43.. . I-...VuV. ”war. -..I......TT-.-.4L~CI.. h...... I..." 4.. t .. ..
._ ._ _ I.. .. I V" ., o
I; 1.1 .1H_..+o»1.¢1 ..V! .1 1.11 11.V .....9I I. lefHIHWwIHHHH o mythnnmooh 1 u
IIInoeu .I. ..V OI+ I .(> $10. 1......r. ......v...“ L110--. “lfifi .0. 71+ O h .fL-OFTII » O
_. 1 .... . n I .6 Olh I . .._.“ I! V .. .
MUHQHHWHunuma. ... Wfo .... 1I1_11...H+OIOI.A, _HHIOOOIOOOFQH..__IHV1 1 +m11fi +0.0“.II 0V
. . V . . a . _ 1 0 0 . o .V..V 0.... I O . IFOI' I.I. I . .V
.. 4...... O 1. V. 1* V a. . _ .. _

. H“”01u“&11 111- #1.! r. H...fwd:..I...fi...-HI010V.O:_... IIO_, VHF Ivy 1 H 0.. .4L ”‘WIOA. AYAA
+1.11...— 1+111 V. .1 .“ 4H~_41M ._. 9 ...»..OIJVI ..V .«HJA IH .. VII... ..ILHInH
1.1.111 111.1111 :M . 4mmGMI I.....lIiIIwoJ 0 .I. ILWIV .o I.. H H 0 00 IWHIPHMu
_1 1 . 1. 4.4I 4 44H _. .1w Humu.11. .I.-£11.. ..w._ w ..I. v :

~w11.11~.11111.1 .I. V A ..V..l«l_;...._49.r. .. . 1.0 .9.0 1...t.... 0. V1 «14.111.111 +1

1.1.1.._111t.11 1 _..1.11 1.1.0 000... t .0 at vi.” ......1. ...1 + +.r.+_V+..1n_vc__ I

1660.10.111 0H 0.. L1 . n .149... .I.. 400 00.1.1014. lwlw .4». . .. .._ .Vruv .a I

Y1O1eof<11 H1_11 1..I.L_ _v- r 1 FLr 9r “v.01. tho... 1IV. v..V0I11-AY rmuwr..9.o.1.0.00|—IW0110I1 II

1111111111H11*.+_ . 1.10.00 .1. 1v.o.I..I..t.r wola.0.on,.o-.01L.0 14%...Hw'col1lo? .11F11 .w 1.1. III

-:::.... . .-efi: _ ... ..:...__....:._;.N .::.....u..”. ..:...U. EFYH

_.h 1. . . “I. 1...V 1< .1 .V... . V .. 10 “r. .
1Hh0hw1munH1FH_I 11F... 10._1..11.Lt.«111111hovo ++fl400r0W1OII . . .01.bp+.fi.wH*.¢1I4rHI
1.. _ _ _ _ ._.. _ .10. I .. t... O‘ur.+. .V.. I I”. h FF
.6...1..J.-_I15V ... . ..: V. .... ._.-F. .1.1 .ka L I -I....Ino 1 . I 4 4 A 4
.I'IOTOQ I.. 0.: W0... IIOA 1'0: I.I'OOI . 1 4.. 1.7..V1l0ll lull .IHI “fit ID. JOIHI 4|. 0 «.:.Iw. . 0.. If 1.1L ”aIMwL +H_ fthfi U
.A 1, . 1. V.. V.. .. . ..IOIV. 9.4 0. ..I 1 II .. ... .. ..I. ... II A . I
HOMHHU1HOHH1.11‘1V1« 0H1._U_OuV 1.1.. .111H_Hi 1111..u1. H.041» .014 .. W.1.I1.I.. .rl O.+.I+..I.1II JIfI . Y.V_Ifi.f.-.‘v01l.l
. _ I I.I..IL .0 .v. 0 ...-0... .11..- II. I.I...JO‘ “V1.00 11.1.1 .l O. I... .1) IOI 1.. _. I.. 4.LI. r I.I'A-OAI

”“hauoonHHMUIH'MI‘o. -“Y 1 a. I 1a. 0111.“. O‘OTuii vhchOJ.._11VI .._. .fltHVQ r 09. _.o‘; 7 110.11 .....I.IIV¢AIA

v..¢..oo 01.01.1111 11 .1.1_.“.10111111—..o.._orfimoa.11.fiuv. u.*_...+0 'AO_.V'HVOVOIIV {.VOGJOvIOHO LflAH.vL

v- OHO‘I.--1001V. , to ._. .. 110+. b 1 0.1.7.. 0 1 r «v I.. A ¢Mn wwblul .vII001JII..I.. 0.0....IJII_.1£I v .1

roH.H_o.¢op..1 . _.V. . ...... _ p.111... 9 1714.-.... ,W11 .I ..I. ..I «1 1 ......1141111 v... 1.1 Mi. 9.?
_ . I . _ ... .. DO ..0.vr . .1001? o ....1.V. ._. 1041, 0V1111.... a I
A of. f..l.- .0 11w... .0: a 4. 4.90 a K . __

fi;;+*w11.mrooo 1V1 _- 1 1.fl 1.1411: «— uo..«.o 111 _. ”01.1.1H1HH #F H a“ W. IIII kw w H
I. _ _ II ‘.1.'O.IIIIOI. 1| I .. .1 I _ 4 4 I 4 0- - .0w- 0tlom». 4. .I.. .fi. I.JW.0

OI mu 1&4 11. 1 _. . 1 ..V... .1 .I. 1 . _. . .11.11.1fi_H_.. Il._ I...II.J..IFI.OOI0....T trunk-1 1.“.4‘IL41II1

FIrow0 ...Horwuowo .0.. 1 0“. 0.0 ...HALIQOoo».o«I0 4-1V00051.__1.?I CPO. __v . HaonfiwuroflTOIY H0 117.17;

1. ._.. .._. T .:.:....g. - .. ..:... ...“ .:.:.“... . ...V. V .V.... TH. $1....Zf .

Ir..o-r.v0o~oufivu v 1.. 11 a 111. .r 091’ 0110.0 ..u I... -....- _ ... 11.»... . w.. vuvo_.ou_oo I I

YIA Otto. 11* 11 1.1 1 _._._ .1 11 .- to...» 10H. 1 1II.OV_c.O.H a m1_1 o 1._.. ..9 .V....»tof... HI I

«0 0.0.9.0021. 0._I 01.00... V. 0: .001”. _.no.wfl1_u V0.0.0I11v1flHHI . _1_ .. .V-Io ..V'V IIm0w1000 ,. 00.»

17‘ I 0.1V . I. r. _ .11 s 1 _ 0.0 n I a. o .0. .. .0 .I Owl? 0..O .:...... H H .1 P I... 4.1. v .10 .. v. 1 10. 0IOWOOL V I ..IOIO.J. .0 V 1.. v't0¢|0.§.+lo TA~IWH

III1111V0 .Ol._..0_.1..-1.. «.1 .1" . 4H.0.-n1 ...11 cw...00 11:0.1-u1orv L. . I _V 4.r1. .|1_._..V I uI_ “.91" ..... .III.0Ywd_1.H> HH

rIvo1cv. 1V4 11 .0 I-. . ._.0V1111. 0111—111 Ir _ _ . 'FI II 1 I00ol IIIIIIOAI .._lblvlrl. 4 44
I>_1l.V...vau_o . .1.I.. I -..-r IIIrnPIL n. v .1. V 1 1 I .I III .11.: I.I-.11.... 4.1 1 .0 -_w I I1-.0...L_0A _.m1 .91 vQIJI

ITAT n“ .1114. ___ ._ _. .01”. 9.. H.I=p .I....L_Ioo,.100 1 .1._ _VvoVoooo H...0. .._ch IMF“... - ._.. . 0 on. r 0.1" 0 . 1.1.w4.# _v0OOLYLI

I I. LVfiL . 0 . 1..» 0 ~1- (9.7.0:..4017. 1-0 ._...I 0 11 o 16*. 0 u I 10 0 1. o 1 - PTOflII 0V$$i+fitvII0oLrI
. . I _ . .. . w . .. 1 . 1.. . . I 1 V _ .V . .

TTOIOLFO .0. .I.} _ I 0 .V.. I i .I.. I I ...-IrIOWIQIOQ4 rfi.“ J... IO "”10 .0 ”._I_11 . O” O 0. 1: 40 I I. |_ Ila-Ax mlllfltnfl.~_$ FWW'OLHI

11.11 .0 . .. ..J- 1 ._ 0 .11 190100_0 V01I44+vs_01 _._ . _r . .3. . .1 1 11 1. .114. 0.00 .10 1 I411... 1. .4_.1.HI .1.d1_.., 00‘ I

ITOII..LI ..a.1 1 1. u _._ .0 .c.r.1_1.w__.o 1 ___.. . .1 ..V 1a 1 .0 p 1 1 u 0.. 0_ .I0v1-oIIo .— 0 01+1.- .011..__..._ “.47.: 1AA

L. .. . .... .. _ .. .. ........V...... ...}... 2.7. ._ . -.. .. n . 1. ._ . .. .... ..I... 1 . .:.:.»...

Y o p I. hr 1 u ”I 0 .O.+_V.Q.1.‘....1_ .0 . .1vI ... H. o. . .V .I .t I 4. rvo c ._w .010 .1. 11w++finvaVOOI I

If. ....I 0.. l . I - I.._1rul VUIVI..._' 11 o _ _I n _ In F _ h. - I. J ._— 1 A! III I. l—V- JQIIIII—. ..I I Iii..hg ”II'IH .Al

. ._ . I U _ 1.. _ II» by b r h 1 .IP D I’ I ’I’

. van a 1..» ._u_ I. .. 1F I. II.I .I.IHI » 0 . 1 ._ _1 I“ ._ r. 0F5II»».0 I.IIIIL.I+I¢-.II1..+ .1.-.”-1++_......#V+4I I
”I... -. .I...__.,...1.... «.11I.“1 1%.... ....14I+ N4I... +.+0I9 1.. 1.11 . -. I4.....« _. ..I . _ . . ...1... 1111:9511

_ H ’3 _ . _ _ 1 . .... H. O. _ 1 _._ _ 11 11+_._. _«VOV1 .AVIH

. .. 1 1. 01 . O .... I n _ V . 1 . 1 1 _. . 1_ a u 1 w "r wa..1-AVAr-LI

I V _ I 1 o o ~_ 1 ... 1 1 _1 _ r 1.0 “u 4.;1 111-1IHV1YY4111OIIII

v.01. ._....-__ I c .1 .11 1 _ 1 o n I u . _._ v o. 4... II-.1_011.I.111% OQIIU

0 ..I. V _ V . V I 11 I.. O

.0. 01...». 11 .1 o u H _ 1 p _1 0.. V_ o- a . I. c . . _ o . 1.. O ..Iohlovh WVHNWIHLI

.1... ...n» . ..I -V.. -..II-I -.I....- .. - . - ,- I . no--. _ .._ _ . . _ .V . . .. .. . .....H IL

.. . .. ... .. :.-..... ..-:T. .. .. _ . . . .V.. .. . L

I. A. ... ,.. ..IVv. . .. .... . u. 4 .. I. IV. I .r _. .17).]! 11V. 1 O, . ._ L 1 . I.II I I V ._ V. .+ IV.” V.. _._ in ....V 1'..IOI {IA

.I '1 . I'lh... .- II I. ’ni..1 I n .__I '0‘! 1 . . _ _ . V 1 1 I o 10 V . 1 u 1 _ AA
. V. j ...- can» _ , ___ ._ .. 1...:
VI. . .. . t. __V . _ . ... _uVI .._ .. . 1 , , V.. . __ I . V . _ _ .. _ V 1 _. . 4. 1
.1 . .... ,V... . .. .. U _ ._ L _ .1 .. .. .... . ..I. V n 0.. ._ . a 1 . a 14 11. +1010 44
I .. 1. . .11... 1 _ _ 1 1 _ . u ..I I . . . V1 . o v .V“ .I .7. 1.9....0IfL-II
... 1 _- _ _ I r V _ 1 _ 1 . . _ V . 0 .. _ . . 1 r... I .0 _:.1w9|v1110.0I991I
.V. In. 1 ...I 1 1 .V. . , ,1 . .n V 1 I V _ V . I .....Id . _, . . .1: 1 o O. L ..h. ItiTVOIJLv... I
_. .. V .I 1 .VI 1. .0 . I.» 1 V a. r . .u. ._ 1 A .111. 1 _ o ‘hqoa tA+¢#IH4i
.1“..._-,. .. . .__ . .. ._ ._... V _._.1 1 1 . 1 . 10.1v1pH
. . . . 4: V. .. . .V _. I . _ . 0 . I I
....u. ....H ..... I. . ... .n . . . .I.-.03?!” V _ _. ..:... .V..I I... ._.I. I. .3 IrIIII omfiip
It - II I plll I. I II I l I I II-||.| [III-I.I

 

 

 

 

 

 

 

 

 

DEPARTMENT OF MATHEMATIC.

M
2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

    
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 
 

 

 

 

 

 

 

 
  

 

 

 

 

 

 

DEPARTM ENT OF MAYHEM ATICI

 

 
   

 

 

 
 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 
   

 

 

 

 
 

 

 

 

 
  

   

 

 

 

 

 

 

 

 

 

 

 

  
  

 

 

 

 
 
 

 

 

 

 

 

 

 

 

 
   
 

 

 

 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 
  

 

 

 

 

 

 

 

 

 

 

 

   

 

 

 

    

  

   

 

  

 

 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 
 

 

 

 

  

 

  

  
  

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 
 

 

    

 

 

 

 

 

 

 

   
 
 
  

     

 

   

 

 

 
 

 

 

 

 

 
  

 

 

  

 

 

 

MICHIGAN STATE COLLEGE

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

 

 

 

 

 

 

 

  

 

 

   

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

   

 

  

 

 

 

 

 

 

 

 

 

 

 

I 4 ‘ ‘14‘ ‘ “{“ ‘1‘ 41 ‘
. . ._ .<I<d.4. I <qu.IIdI.< 4. . .«« I «I.I.-.I 04¢..T
1 4 11 I .1 4. A. «41 I. . 4. 1.. . . .. . ... .I . . . _. _ . n. .0 o .._, _.l 01......" .. a. uvooj
1. ~14 1 , A tIHIHQHJ< “NI“IIH—M IHO [HI .IHdM 'HIH It‘fifiofi IHfiMm «a... I‘L..?'OIHIHLVHI [IP'IlJI ”.I..-.Olfi‘lr _. . . . I . , p o ”1 H. a}.- I.. I. I ,C‘lIII I. ..I" rI-r
IYAVVQLJL 100'. |_ 044 .. I.. or.‘ I 4 0 4>4 I 4440:.” IA 0+04III4I'IIOJ I.v_rI.Il1'_ O I.HIQ. . IO. 0.. ., U , . I . .I ..F. VI I ._I-.. _ «Ia. II,II..«II I: I4 I 0.1!.eu
r+va9A *IaT. .0.... ..LI. I.....I a I. «.4 1 « ?> 0M1 . Ibflp+flAvlvlToao1vlllAHi...vo 1: 9L .0. ._Q 00.. OH. I. I ..II II ... ov..l_.|I KL...IY 9I_ III w... W. I.IO.
rrIAYvIQA OQOIQI .I...L-... I _.IaoYIdI «.._o_IH¢.A VQIIAOI I o ova AIIQI_QNH Ido?...990+.wa. I- v04. Am I ".LIIA. I o. _IHLL I .I- ..tl II-_o___..00. . ..II.1.I.I4IIII..I.LOA
IVA vv 0L4; ..:-._.IYTIIIA _IQ._-II.._II I coco.¢.a ..I. «9:4 9* AIHIOATAILLIsi. c....:.. a I... ._I..*o : FAA... I .+I+lIII I... vIII I+ooIII I. - w. ..I ....o ...9. 40...!
It: 'fIVALOIwI IIIMIdO flu- 46.1 c I. o... I. Too-IIIIOII oIol IIIIIIIO 14...!4 I II 1 4H4++019c vamnuono¢w LI” 4.. “.m. II I ILIIIv¢|1_~IO.dI‘Iv.l. ... n ..I L. _ IL _ co-.. .0. ..IOIOITTIOIOI
I . . _ o I o o I.I. I . I ._ .....I. u l
-r...r.--1:lt...;r.: .:.:.... ..:.2; :.a.t.nfiur. w . . w. LELEII ..: :L...:....._ .-.I+I..NI 1-...--4 i..-..1f.44...-...+tt19
._.III..WIIIITII.II . .__ _.... ”FaLIoUVHHIHOPOObJ.ToHIHL. . II III L- LILY... .o. I..I ..TIM. I... T. IIIIIT . IL I: IQLI.,II_ I -Ar.fl...o.r1¢.. 03%.. ”.4?!
IVA m I o A. L HI P o .. 4 or ..I... .I.”...I I..I.fllllolu ? V II F H. 1 .._ L14. >I|>1 HJT _ I _ _I‘il... [If . 9 “..II I .. . m . I 9 r4.
> .11 . III-I , ..I-1.1.0. I..I}. v 1 4 I. H w _ I 1 MI: I. «I» . ...IIIQI. l IIIIIIVIWM* _ 41.1. .0101 I I f o ,. I w 4 H PM
11 L L". .I we re 1;...ch “ . 0.191....1-‘ .Io..c..¢|.l -I. 70.0.?le I 4H I» I 6.6.-r0 - r..:101a IIIII _ TAYIAanJIIWIIIIJIr. ._OILA : urafi I A
194 YQQIIIII .IAII: AL... :0 o _o O I I.IF. olr .IOIIA . “I. .1 11“ DIHIH.<IOIIAVOIIOAYOIQYIO IIOIII'O.OVI I0... I ”9*..‘IHI. . IL .._ ._OOI’ISOY-A
A! olIoOOOFOOIA .6 ..V. 4i IoOronooI' .111 41"}. TIP'F _I_I OI 10...- 1_I..1. :I'ITI Tr I.I. RIO I .Ib§.0rll
..I... ......II“ o _ II ...LIHII I.I4.Io.I.'1d 5. 1 $90.9... no fa? 4 IF 1‘11!le 9.0.1 OLIaI I I. II I.I.I I... ILIIAA
uA V.. ...-o FooII QYOL ,ni. ...“ Ho. oiIH bIrD-rD .0? ..II 1 ..I; ‘IINIOIIfIIOOJO om .AI .0 at. Anvob 91W;
IY AIAVQIIIO II I IOIH” IIIH_._oIcIIIIAIIHh4fiI044IHLI1OHYATAVIOIQIYv7IArLLwIOOLIFt 3.7... L Ifio ”I’LL 0 ”flu-HHHILQIF A. 9+. .4 II. MowoooT
I.I .I.. II II I_I . p o .I o... co. v. I , . _ 1
11:10.9?I'obOIovbI05 L.H__o._.IN_.o+A“I_..LH ”H” HULLWruwouLAV Wonloofiwooo .+ 400 If. IWI.—r.o L I I.” 4AA IWAATAWA
-I-».....-.I---..II .... KL . I- M f 111 I f“... m i “.2.“ u: 1 : «KW fr
0 1 4.11 1 I I 14 II I<II< 4 I I VI I.I —O OviOo‘I to _ _ 4>1
XXII! VIIIOOHI’Ifi‘filfloI ..II I- IH I O L14HI.44HAITAIIHHIHIJII.II+ HIOIILLHIOIL FAA “Hr? 4W _firv.+?I.QIOATO I O _H A o I of». O. ..I .._. 4 _TA rF H V I .I ._I... LVN.”
IIIIO‘ O QI «.5111 FIOIOIOIC I.I-..Q‘ OI’IIHICLIwleHIH “PHI... 4.4... I..! Ifi‘ N I .r4 1. b (l1 .r .70 IV II D D _I Lr ..I”... l D... Iw-fll ..V. o 0 ,- .¢.0¢ .I. I I V.
IIIOarIIIIIwI “970 I.” -.II-In“- o- OILI ”HHH1M.ITATAY.¢I?IAI o++¢~royv 4.... a ....ooco. L.... N I_ H . . .I..4_..o.fl HIInIQIIAI
I.I...IIIO ..I.)} I. I I I .I... a I_ . DFFI b bL I? D 1!?! I II»! _ ., H . I. I O IAYA
. I . v OI... ' 4 14.. _ _ I . I . I w ..I... ..I‘II
.....T-.._.i. .. Antwan”.-er .:E . :7 .:..-; .:...: _.. n; . i m. .. I -.-.
P1. OI A I..? | A 4.0.9.». 00%. aehs I o.v4o ..II. .I.. _ ..II.‘ .4. 01...]..004
rt-.. , . L . £11.... .I. -_I-.. .: .... . _ 14....-- I.....r
L Q. C I. _ I , PP b O.+.o h ObOPIVIOO DIVIIIOO’U .. ..IIOIIIIIQI ..II-I. I.) _ ._ film”
J4 »H _H I r .— IICI‘ I1 ITIIII.I.I[ HI. I I _HIOH
I . , l I .1‘ . . I
H¥1HH< _ 4 4 I fii 4 .ji . o I. Iyo. no. I I 000. 4 L a ..I IrIIl '1 . 4 OAUI
4 I11. ..§»..h. a... _I ..o :.. ...“; I. I L....I o A . . I... or III. -.I..- v4-1.3. I
>06.I.IIA r- I. .0... 4....er * I.I oc—QOO .V. 0* I .*00004 _ 4 ~I¢¢.wuoo IL_ t . _... .I.}..lI
YAAM .oqu .9. __ I” conov ... LILI voouo. ..II Ac. _ ._._. W .. I91, . ..OI ...-III
H _ v7_.‘000099 A Av.br¢v # o I o: A 9961,4110 .— I?I¢IO‘Q .1. w . I..... _I
110.4I _a I... _oL $.Hf44IIHTIYIIQIVII111. .471 .111». I.“ . .IAfiHAIIILI..II,kr—vf. . IIIOL .v .InIH»
YA Av.6_1 o I 4. LI. ..IIOI 1 9+ Ar: ._ O 9 ..fF 4‘ I. 0 HF v." ATAIIOIO O. a. ..I v I IIIIHIO I u: k! 9 r. '6‘IOIOI9II 4 0 .. n. + 9 . 1., I . O. I 4
11137.43 ..:....;......HI. 1......6IIIIooI. 4+. .._.-II ...Ia....__._;.,...... .f L ,. -._.-LIA I I I-
[11.1.2 1 ...- ..:.-.t... ..S ..III L- 21.1.... .3 . .-_, I ..II, . .. U . .. .. .. ..
rIv04+4.IOL+A 0 IDIOOIDO. 0 Lo. _.4I_ro.-...+.o LILAkfla 0.0.0044. OIHIM. ._ .. ... .c r ......I. I ”II IH . w >I 1
.Lrjiii. .....-r..:???IfItf ...: . .T..n:. .. .IFFI I .:.:.... I.I..Ifln .. a; . T. .. -I....nu
IIIIHWIIH . IP11 I A ..III I I II I 14 I HIIHI .. 0.1 II. #EIAILJLIIIIIIIrLITLr.Iv... __ Imp . .._ ___ .HIIII .. . --.L-.III... «I.- .. _..; oo #1
"1.1 o ...WchowI94WNIHOIlI ..6....~WIQ. 1 4H ’LII‘.III IUY,‘_oftoo..LaOIuo++.+I.._T ..I.l.... . . ”I....u__+. WHW “W H. __ __ _IIHIII.I .IIIHcILHOoHJMIH ..I“ 90...|%
v . .. II ., . I.It xvi... 1|. . ..:. -I.“, . . 1 _ _ I
If... -... u.....?3:+;.L I41; I.I...I... .-.-..T....-..+{.....I....I . .. .. .:.: . H . ..-- .. : .:.:.: ....I...~.I--.
“YA HI. _.. ccA ."II _ I .I. L .._ I o. o I H ”+7 . __ _ .. I I 0 o .- ._owmt-.. I Iv...|_...«lv._ .OHQFLI. i _LWpILfLLIOIQ’ _..III_I IWAJ
YA o. «(I.. I. ..cIII_,.IIIo+o.OI *c. §. :1 11 .1. «Lr’fl ..olir. ..IOIOOOIAQJIA ”II 2.-.}.-. |1.I|.['.II<I.I'H.I ..II 9 .0.. _ ... o.‘wo. .1LIQATATAI
If .‘-I-. I __I .:.--..L fir. OII.“.LIO’bt ..I .u _‘I .I....Law .1... ,9. II I..I .qu..I.OI. IL ..I on... nmkoovobl....HIvlfld..IF<
H.4w—II. 7N! HHI"H* 0"; L . 0+1..#QI.OIOIAYAVI¢_.O.OI_I 0.107... O. I.I.OIIA _I II. oco 47......“ I. 1.0II ...I. Oliulilmr IDI'IIIIOIF-Lotv. IOIA
11.11». II._I.I._D ._..lr _II.»%I..H. .IIL.II V1 . +3..va I‘QIIIIJIAI*¢I I 00A . IVI' .AIIIDI .I. I9 ._ . I1 .1 I..I ..Iww
III A 4 L .0 o N D .... I... .IOIFI I. ..I.» $.4I. ¢.+H.Ib1 0 1 #3 I 1W1... 1.134-001. .iI.Q 1m¢A II“ '0“ I O H. ..I. u..l(.l..vlo.|0\‘ ”J _ _1l' DIV HF. II ‘ l 0 I IOuOuOII 91ft!” ' I vAv
YAYA AIoo. .. .. r «14 .— «u 1114 4 «I I _ IFIIII .A .. I IInolII av. II‘II III III! -15..
YVIA I “I.. . ouwfl IIAAFOI Iawfi 0 ..Lo II «I'LuLI I.. fl\fil on 4.0 III ..I J7 ._ I _ I 01-0.... .0 4.111
vIvAIoo I” ..I. Lafitn I- .On...OIO.QJ0n.IOI&.4.I'I ._IIIIII ‘1 +. . .4I. _._I 1 Lot. IPOIOIvI VIOHWJII... .HHHI ._I-..IA. I I .0 I.-. IO'OIIUIA'I'IAYA
I 1IIfa.o_4 F41. 4.......fi.fl,o... Yo»... o.ovgo.. AL. I 9 __HIIIa Mk. II _ovIoIICIIUIIIOAI.I.II...IIIIIAI
TOIIOLo on. ... ..I. O > 00...... yfiahOIIII o YIOIw§OfiWaIm§ L... I. .._. UW.I . ”In TcoA ..I I.. d. .I. _ I.I . ..II.I.IvAvI.II
Ipbo.o#.o ......ILL..IIII YHA.‘ .._.” ......cooc 0 I94... _._ 1-4,].-. ..:. . f ._I_. _ +_. .I.-J s...o..~14-o I.I:
A+ I....IJW I.I_-o0.9.-,IOI Lhoaa. I ..rttboo. +4.0“$»- o... fl... o .I. _.H... .QHQIII. v I. I I II. I II- .._vlaljllo... ATIV A

AVLVHHOIHIAHH.I _I.I_. IIIIIJIQ .4.I~0I4h1. ouo.Ioo H4I¢L+ .H .hu .mIH .I. .H . H4 .I I..“.hIm. .7 .fiw}. . I..I-.r_._.o.. Itwma. ....6..w...m|_

I _ . I . I I . I.I 0.0.: .N . , , _ _ . . _ . . I... .I It...) I. IL’I I I I
Pitt-..“ .:...... .I .:......u..- ifhfihtII - IL _I _ . . .. I _ ... - 4 - f ..IIHIIiI ..- .... ._.. .. .- _... -..-..f -... .J. ._ .L 1-.
P I 0. [I.. , . I b ...} L I 1 11.4. 4414 1 IH441 I.” 1 . V ._IIQO . I..I-If. ..Ilr 1....-. ... .. .I ._. a. . .. II.I_|I;_. _II .I... ..II_V.&I.I.\ O52: ._ I
_ .:4_ Gd H4 JHHIH HHHW M 0 H « I.IVIA rOIOI.IOH1_Oo. OI.OOIO4I_OWII._1L.II4.QWHOA¢ WHHH,JHH .IEIII 11 L .I .OIYI1_(JI..0..¢I0..IIII.. Irmy A... .II ..I. — .Ifi‘fIFI “a 1
9.0.4.; I O..II¢I§I.I - _ A. {I “d'rct L... .A O v.7 H.5I. ..n .tl' il. _fuIJJIII‘O. W . r: .. o 1 a}! 1+LUO.-. .l.,.+ '51:.‘HI.I..IIJOI.. I .I. _I .I.... _Iifllu _.m . A

1 ooa .I— 0.... . H »h , .It 00- I.. .0 .00 I T . II A . vL +0. ..I .I...-III I Iv. I * .

Il+AvooobIIA . -_I._.III40 .o 0. 0+ “«¢.ewvo.. ..A_O.O. ALAVI IOHHIH ._.rIuI A. .09 0.05 a? ..II. X "I a III4_ A WLI TOOL A IIIBIHHWIJIIIIHIIHU ItIO‘I‘I ...II .I. _v. .0 ..Lifl‘wr L
IUUOYANuHUo ” L,.H VI ..IIAHO4H «.._owQOoL .HIHHAHAv. .V.?I..r.hfi-flt LI! IOIIIOH ._._v+++I_#v "H“ .mO. ..I. M11» 1h. I.I...HIIIQQMHoVIIIIIIL Il_II_fIO_o_v II N how
1 v . , . .. . OVA. .OOOI o ‘ I4 I. . I..- x A. r . .IILL .. 4. ..I. . I II 0.1.! 0 . YA
IIIIA.4..OI0A.I ”on”. "HI. ”HAHN 4.“...W_ ~oc I...” _II ..II; .“Ha “do LIOIII Lot. _II? I! In mup+ I . _w ._I _h‘» ILIrle‘oIIL; 14.“...4- “:.Iwcro I
I..I. v.0 ., OQLI 0740I. I. nto Ooioh AH... _MQ fl .IWLI . _,o.. . . .._ .V. _ o... F. ,NMI— III-«ooOon. I-.IVIIIIGO..OI v: IIOOVIL
III: 94.1-... I I o .1 II... . ”..II. a _o..H,_¢I +549 ..IHHAHAfiTTO : r ._«r I .4096I _ I.. a. _ _
.I... o Ina“ o I; LIOH IR... . .V...IOAvIA OIL,OIO._E*IL AVOLOOA F L” rm . .u a“ I"! I'll-IIIQI_O.1I1 {14.41
TA Iv*.r4. _ _ . . II... _ L... “If 1H m»% L _1 I.. 4 L I «HII v. H. I . .IH._ LAVAIOIYHHKH .IIII.
III _II . . . _ III...II.. I.I..I. . . IrI._II.§H.II »4 ..II I..IOIQHJILII w «Iw. ..IIIV¢.Y§V I ..: . .I .o I.I.._I. ._.. IoIvaioIItLAIvI

It... I..I . . u . H. , ...-..II‘FII . *LII IIIIIIJI¢_-II.II_,I.IIII .I. <4 $I9+AI§I0904 LIILI Av. 4+ . o .u o v V.. v . _ E ....o .I .0 90.009 I..III" .11 I. I ..I. IF I «LI
III L. +III_IIIA . I... or... “LII. o II..._.I0I. ..IILH. I _1 I01A1'00IrIOLIm4A .M.” fobx ‘. Lt. _thvI..I_ _rI..I.I OI... ”5.00. .I.» 4 AvAIO v11
IrAA vooowIA. IOI .00 or. 21". IOIOIFO It‘ll! .-.‘LILOII.IAIIOIOIHQ“JQ.H « IN» I... cavt. I. Bib§04_.o.r _Lr -IAVO. ALIA;
{VAIYOOIIIY.IOI’ITOI4I ..I. 00 Irr‘loLI..I....'..'IV,I_L.IIJ_..I.IQ.H.IO v5.9"..{41 44 1 I 0 +5; 4.. _. 1 I IflQrIL IQII .Ifl.’ so”. I 0A A *o YAIAA
179999 >” o- . ., . ..rL_I¢.ooIIo ..IILII.III¢A.IooaAaoo VOPTOITIIITIIJ. r5064 ,- Io LI. .L..Fa III I .I _ at A11

.. I ..II II. II- .I II LIHI 40.10.0031? II+J$I11+IIQ+IO¢I¢fVI . .I‘ 07‘ I IIIIIITWITIII a I I .46. _. . . I TY. I 4 A... . I . _ILI I L. .AI
Iv Av15.1..%[. .r o 4".» 0 v .I.! b Y”..._.¢.V¢OOIIA ICOOLLtv t..406HI,..4IAIIIIOII4IOI _ “I .w. ILrI... .. IL I.I.TqIII.._rI.*_.I;, « ..v H I 4

I Akfto'II . . I a .I.. _ 4 . I ht.-rI +r114¢IIAHIV c09v 0-7 I»? ,II ..II. II Lt‘nv.ll,tn-lIli.Ii4||lu-II.IHL .l lnl .IO‘IIOI III (1104.. AA

h Fr I90; I .V. r\.“. ..-IIY..IOIIVV ...fllft 0930 .0 WW.— ..I > I r I q i r» I * ..eII.I IIflI‘IIIfOI.I..II%I IHIIIIV. -IJ I. I+I L. I . I.. .I o o 0* d . v. AK

11 IV b 11‘ 4 1 11 J11 1 1.1. .. u .. ..II. I. I _ . .
H H 4 4 _ 4 .1l1. 114v“ .41.:1.ML<II4HN HIHII.III’OI.. I N o. H 4 . OIAIHI k+h1.:1r. 3.m“|rIfiIA LI .. .. . . H , I.I. .I. 1 ILL .9 .AI. 0. FIEQIII .- I.... ..I.._I.O. c A h. _ a t b u . O o m I v H F 1. II

.- HH MHH l I. _ . *e I v vI1_ TOIHIOFI_L£ L . _ I ., . . _ 60.0 0.0.; - I.L.w O...'!I OI. OI... I III. - . I I I .. .L no I u H.“ I u o t v. EAIIQILYA
1Y0. NH HHH rv «I.?18.¢».8.. . . 104584 I.....fo. .0.. I..«.I.._. _.. a.” I. 7901+..oc ..AA
TAYLAv<1 4<I< o IOwwI>Ir>_ 11 TIIOIVLI- I.IIIIIoQIII. I...III.,I_I ._ m .I. I ..I .9900 III

*I 0...... L. ._v.?o.o$__l._r1§_~1v . , ... I.oY.~.I Yo.IA.IHIlI
1101.0...4 i‘éerQI or ..k I . o . ..IAYIVIJ
Iv .AvOI0L4 _HFAIL Q.l.l. .4 I... , lituodqfllfia I .00.. «.0.... la OI. diilt. m. D‘flItH H
ILIYOO’fifiIVI. LL L ..II... ovovuooumoofna I. It I00.04T1 ...:I.HI_I I. b LL“

II VIII-.09 IIIIOILII-§_ [LID _II 05. .l I-.I..ol I..I-I I14 I.I... 1“ «I. A
II +9. I. _..e .. MM A. w .0.?o114ll... I. IIIII..IHIIIII[I_¢..-L_:..I..4 . .... H I Wob .w

r} I h _.. . ..IIIIH.4. I. o. o. «. o?.a.IbI_-4Iu.IIIJ._Hh1v I¢Y+¢+IT¢AIAY
IIVAVQIOIO I..IIO. 4 o I It“... I. F ILIIOOOIf I.I.IOIOIII: ,‘VfiflciLO0ILO VIVIIJJva. LYQAII

«In 9.0.1 .. .. II. I. LLTIIfIvoIT...’ I .I.. I4...'I—o.aa I. .1_9IAvA ._I.YIHIH I A1
ITAfi‘. o c _.M TUIIOLFIO+1I 0v.0..¢.o: Yd II _ .II I a .OQI‘ v1 rTVflL c I
11.4”.«OI- . OI'IAIAvrIO v4.11
111...??? ..I .. .._ _..I. W .4 VHIIU
IIV¢¢OI0¢00|I > F..... Geov‘. ...v Avl

b. b L.’ .

1“ 44 ‘J. 1‘ if b -_D If?
IIv0.oIonoo M.I.A.H. A'IYI’UD'IO'I % LHL
114‘... _‘ 4. F.. O O _.. I I I I ...?OI..I..WII..I I H

_ HI... FIOIfO 101141 11.»— I
[HIQOOOOJ O I I..IUOI 4OI¢.#|OI9I+IVI4§llII.IrI|40 v L
I IOIIQII.._. _- ¢ IIIIIIIQILI IOIQ‘+OI_.$.o¢oo.~.o VI
I.A:...AI . .. wood; I-“ 4-004.1Io.¢_.4.jé‘ A
IIOIPIL¢5 H I. ¢¢ _.O.,. a...» 4... vaIt
I o.“ L I ..III Ir..e._. .._Ie. ”__._..voorIoL
r , . n I I II—r- ..flu.‘ +_+*I .*Fkh..rlbl.i.wt
I . . _ . I I I I..-.141. f. I..I-tll
TIA . ”I I..-.I..- "I- ”alto... 4.1.VIHI ~.I\ .IV'OiAI
I 4 o v I.. 1+.IOOI... _I01IIIoIfIIvIHIrov .0 A
.. . I II IPII t I I._I~..I ..IIIIAY
YA _ I.Io I III. I «.4.. o .IIOIOI .OAT
I ... I. «cum ». I .0 ov- o.
T. . I a I o «u. 900 .OIOIIIIO
I a o .. I I I I _. .. IIIOIIII I I III _.
h; h. I O. . . .II .I. .IIIIIJOI 'OOOIL
A W: .I. .I O I IIVIIII. I
I . I on :. L I I I... ..
I ..m _._. »4AI.. I , I. I..-LL~I+I
I IFllbb. l .0.... I.I" *.$'.‘ 1
I..n. .A . I a o .I.. 7...... .... ... «ii
I... . II .I. II. v 0"..OIIOLO4LILIL I
I. O. I. .0 l I VII 0| VI 31.. TI’ITIJHJI
I L I I o w I I II I ..IIOIOOQIIQ‘ITQ . Al
I I I _ _ v p I II I" IIIIIIlrdIII .:...?i 4A
I I _ I . a. _ I . Co... .I.. .I.. L 0L1
I I.. u I _ 4 . . c o ‘01?! I I I IVIIVIICIIIIIOI 4< b
I I I . _ h
H IOIIII I I II. II II I Q .I ..I.- II sl. 1... I.I-tnrllflongl I.II H44»
\ It I l I I r I II. o Ivlc-nv..rv1:|70|.l ll. .— 4E
_ a I I _ 0 I 0 F.. 11‘. 0| .1 14344
I I I I . II ..I. I III .I II ..I. W. 90 oIvIbIDIFI .II‘HHIJIHAI
U I» I. I C o o .0. 00-. 0'010i0Iv‘I . 1
. I .I I p .I I. . I. I o_ I.. ” IIIIQO'IO Lt... Y._QIOYOI#J.I¢.OIO‘ II
.. . I II I . I .. . .. , b. .79.???91 Vivi
J . .I : d .IA 0.0.0 o I. no... JOOO._¢. V.’ _ H
II OILII,.' IIIO.‘ _I ...I T I; IFIO .t n I It." .I I _ a... ..II. .. | .I II. III“... It... 0.6 _o ".IIII.0_§.+._IWII_ M..:*1 fivvi11 H
.I. .IIIAI. It... IvIoI ._fi .III. I... I. ,o .o‘. ._ . I . I . III. I. ; III... -IIIIIIIIL C¢§ .9. OIHH Wi1HH
.:...... I1.9-.......... .... -L .I.- I .. .... T ., .-.I .... _, .... H u . .::....-I. WI. .uI I.IHT. 4H3:
0 6.: I. no ..I. .I ..I... I....vOIQ IIr ._ HI I? .I P D D I Ft» > r b
....nmi 37.7.. F ”-... $122»... -

 

 

 

 

 

 

MICHIGAN STATE COLLEGE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 
    

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

     

 

     

 

 

 

          
                    

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

     

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

                                 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

              

 

 

 

 

 

 

 

  

 

 

 

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

              

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

    

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

  

 

 

 

 

 

 

 

  

 

  

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

    

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

 

 

 

 

 

 

 
  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

11 1 1 41 4 4 <1 1 1 1 14 4 I.I 11 4 . <|1|I 1 11<1 1
>1 H 4 4 0HIHHrIH. o F I a y I § 90 I OIA II I 1 1 m H ‘ No. I c 9 a OWH > H I. JIM “L I. 1 I11. .0! I I. I l I 14
fliefii. . . v.1? . .-I.1I.. III I .... Am.‘. .:.....AAfi IIIHH q...... u. _ .I 1...... J?.
AVA Azow 1 . . 1.1191 . . o. IAIIIHIV 7AA: AA.H I A. . ..II... -JIHIH F.....«A _ . V” ......t.............
1 w L 4 ..II ..IIOI Al 1|0 III A a at. M 4 .+o-W_ _Iifl [‘1' .0 V%.0F.:<Ifi90
I I I Qt“..r.o....n. . . H .vA +IAV+LI AVYI . 0 IN I I WY? 4..¢IOA4A_ V_ _.1 _ v I.II4 ”I V... uao'ofl
% III AvtvoTATYIA +AO$IJ+IOIOIII ..II 1."-3. OI v ...V ...» Mn. N .o TOQI4-I0a‘Io vavIaro u. .. ......+.«.. .A _OIFI. .9 I..... -' J'.11_0_V+«4
. 0 HW >1?>>0',Iou ..II- ” W1WI IV ..II‘AvIIIII'IOIIdIJ‘IvIII.‘ I .46. V“ H a.“ ,Tfi « I .HJIUII'IAI -I. I_00 H... A?
4 b ~+¢<.voouo IVlI, » I 1¢rI_.IIL.OH .O‘IOf0501. AOcoo..A-.r H .... v ..v... I IOIIOIQ . _MIYJJW .1...
0 1 1|11- IIIIn I..I... OOVW o +HVw coma-F.. .¢...oo IIIVIVIUI'IILIIQIOJIA.IAYO_4.Q. 1.. .. ._..QI H VI VII . flu] I1L_ _ a..*
I» II 1 1JI ”.11 *QHIL..PII’I.IIOI?H.OI ‘Il _ ....II .IA, w# ._4 I'IAIIUI I. 1 .. a. [Q I_tI¢Io ._ a o
. .HAIO . . A . o . . vac . rt u v V..? ..I .LIILIO I.I.filvl I.I l1. +4nflo _I .1 H“.- , .I.}...Awlid O
4 11.! 1 0 Ilfi ALIA .nc .. wad uO‘vQOO o tIOquIOILOH“ ..Hlmei ..HVa ..IIIuII I XJIIIIW V
3 1,6 OIAwIAWAvIA I’oI VATATOAV #4 ¢.o¢ o o v a. o . . (I0. I IVIIIVFIJ i.r+fi.f.r . v 664 .II.
Av T YA A v 0A Ar TA IA I Y A ALTAYAV v a ¢ a A o . o u . o u s c 1 - a H . o b c . u A. Q I IFI I.IVIIInl .0 #1. OH. L I I..I .:- H a. I D 'II. I.
YAAAH 41:17 ATATAfi... 7...... .... . .F _“.u . y. ..9 olfl VIII-.44. .IOIoIAYv _. v.4... . "OJOIfiIOIQJIAW‘
1 14.0 111.009? «0.9 H. V_ .:...fi. . .fi-obvv..M.AIOIVIII 5.. l..wc h . ., : O VIA-3:04
V IlflII. + 4 . IrT. 9 .- Onvoffl.-r. A. __.uI-H .:.:. V a v.9 05.91,.
IATA TAv CA I 5 TAYA A *3.“ M I I +1“ + IiIOIIAIIOI .IJI1.*L_ I a .O 9 c 4 .“f ._ I w a I It A . a. H _ o ._ . . I. 1 JILHJI «
I . A? H 014 NH J. ..I. IIIOLVII .....L. III]. I ..:.v I... .0.“... role A_ - _... . rI_Irf I O v1.0.9.9.9 Ia
r I I I 1 0‘0 1|I1 1 04 1 r1<¢1II “ IIILIJIIIIdn1I.IJ-'n.....iu.rl>1 1 vr . _Ilal _ 7.1.3 I“ IIAI. a ... .::.IoI4riIT4I.
O I DID: I.IIII.IO .I'IIIOIl-o - I _ 9% A. anL. V ow - u A] _. A ,0 n o .. u '4. J A.. _ V.. _II. J. A 1.1.. . V
0 “I .9. ’4IOI'QIF100FLIIIv. Iroo+_ v«._o ..flvV . o:4A.».—J MMIJI. I If I..-1 r...»— . ..
A ovAY Avv I; I; VIAlv I4. .99... . 9..” .fi. ... A. .f.w_m coo .L *LIIIIO-Ioil, _IQ I+I+§IH_I._ “I .I.. ”..IIoa . .+IH-H . .
1 1 .P 'I_I. I «It '1'... IIQI_+III..L.IAJJ -1;.4'AYI+VM' Ho. 0.. - Hm A oOVAOV1 » Av _._..oalnva J .
I1 10A v .1. .1 v4.0... +A49 a“ ... vaooovu .v ...vb....L I_, ..UJI-I...V.,I0AI. Q I. 5 ..II.
1V . +1 I I II? ¢I¢ 4I'I.,. .. I. If 0 0.. ..T’O i t o ATEA O . u o a no fin v I 0 u 9 HI . f. ..IA+0 p .w c, _ A . _ AI. 1
0 .I4III Hfl+ .IJI I954I¢+4IIOIIOIII ..III..,I.«4IA.4I fixwovcHn. vv..os.oV,0Aav .I.* . 1|.w pco.._.t_UI
a _ « _., o oA_- v.010‘..w¢lao I_ o ... V.. I.. LIA VIAV. J iaiArA .l_ # thA _ ...Ioo
- I..I...F .HHIII v.V...d|_ VMFIIehrro .0 ItoAv 9,.vc-o. 96.0I..I+.JYA_.I_.I..+ I.I..IIII _II 1...,I_L.A...A
I m I .I. a. w _ HI I . A». 4 I I ,rtmT.'dIoI....i.Io. I-I_+o1..l . .V.. .. . L I_ .If. LI
1 I ' [VI-II ._.|v<. .__ululfi 444 I.IV QJII OI... _ .17 VIIII‘TII wII II r I.. w ¢J1Ifi .0 .1 .. o o A A “I" ._ _ m. _ _ _ I n I I ... 0 ._ . .I. I. J. .
IIAv Tr L I "*OI OIIIIY1I0III4 .L. .I.. IAIIIKIOLIAIQIN , . LTvi .0.} ,u v I. o ,0 _r _ F _ _ km IT I.I _c «f . .Au J “I..I
ITAYA: T: 0v oAVAv 1 .IoII. IfiIluIaI—I... J.w..V.v_H-IV1H.1-+II . A. 717+ + co « .1 _. +9” or? .. .._ .va . .r..._I.I
TAYAI AI 0 v V. VI v :TAA A AIAYATAVAYAIA. .-1. I-.. .I.:QL . .I. I... _fl ...+II4 . FAIIOJATA ... . V. _.. ...H < .I. a 9.3 u . .,,, _._. .II . .A _II
I —r do. o..I._. . OI..I. .T4+6 ._o' . .A h ...- v burfrll I.I fl. r... .I.....¢..I.91
— 44 h. _ H 4 . _ I o. .. i
ITAvAvAV A A AA IO . Io... .7 I ..I-IraI I- . .4. _I. -4 I 4. r. I ._ I v - i. _ _ q . I
___.-.....I.-...I_II...._...EAL. : .. _..; . .._ .... --
I o I ._ ._J .. a... I-I-I..I ....I+LI.+J.o.I+r._I+...ITI_IL 1.....- .. A. ._ . . A . . v ”tn c. H. . . II. - A
(J-.. .I...IIIOTIJI.I9LI.IJ.§I.L,IIIJ," .. . __rI V.. M.“ . . . I .II ..1 V . ._. .III _In -I A
1 III II.II.. II'I'II'lIIdH .¢.ldl.l.li_ll1lrlll_ A «III_. , .l'l _ L. . I F H. l1. . r .. I
II A . 7 Av v . ._. . . . _ v o rIIT..8 To- IkaWf L _o.u_I-I.....nIJ ..IQrI JIV. p. ... I . .. .._. .. u.. p.”
I 4 - 4 .. .::. ._. .I.. 3 ..-....I ....I .. . .....-I
o o o A o _O _ _ , A - . w I 0‘ .I. I. «I. a W.tlm.l .I.... 4.41 ..I. ._,..OrI . I I I 1 I
TATAvLIv .Av 9’9 ‘v i ._I b ...-I ...V IIOIIIIOLJ 1F). 0.. J. ..u..¢I .V. .V ._ -. Iv. - _..I.. II
TATAvIA Av I.I} "III _.vJI obObboIIJFIWIIYOIwJI 4H; 4.1 T- A o. I... o~ _ ._..IIIIPILIO .I.”
1 1 #II I‘VE. I .. I ..II .I....du f. 4 “AV “J.I J .I. II ', _ k
1 vAA I‘.§IA$Q QOflI ._o_ ” _01004.‘.. ._.-.IA. «1 .a. $Q» .LI' I .II . .1 I.V_ I II ..I. _... . I"
A Av. 9'. 0AA+¢Q # a a . VVO.,A_VM.AI_.._yI-_ It +oo . . o ‘IH._.V..V _ ...Hfl V
IIv AAv . . vooOYOIiA} . o _AthI-?00IIIJI.<.»JI.IJI~. «LAF n......... .9 ... «._Fi _..1..Il.w_‘# .V.- . M .0.
41 01,11 101. n .wIIIIJ 5. I [JIOhVIlIIIIIl II I»! L .1. D IlhIII 1 _ .. L . .. .... _ AI! _.. ._I_._ I... A a v I. .1
YA vAvAIIA AA. 1.1 1 II. .I. I I Ia IOWA]. ..II. ? .AJHHHJJ [4... H ..HI .AIIII a .0 Ihr._..'l.l.._no?I . o. ._IIJIA I . I ' VIII—H I. I... .IfIro m NIH
0 IAYA o ......oo o. - I..I-na.IOotI~..II¢o. x H U I ,.. _ A IV . A .I I10: .I .I4... I. * . ._.», n o H
V _ A_ . + . In ‘ .OJII .11 V "It [_II . r I I I. _ A _ __ HH
IILA I I I I «In I 1-.-..- .... II. 1+ .. I_ ..I _o.. . _ .. _.u _ ._....4111
I 1 I I Ital. ._ ”J In VJ _II. ._. ._ $.I.I I.IJ [I.I IAIII-III u . «A1... _oO...’
IV AA v..AV . . v v 19A rYOoH I . I-f . . . 0JJI0I I _ .o no _‘a. I
1 1 1|? .IA IQIIOI - .... A. u _r _ .r. ._. .311
o 0 IIIVr . o . n“. I_V0I90.II.I.OQ .__wvoI
.0.-- . ... ... .... . . .....-_...-r
LN V .... - ...._ -.I. ..., ...V.. ..II-.. r
I Ll. I I.. l, I.LV . .I Ajfi. V .I I IV V .0
a .A n 1.15 .-vII.rIIII+IV7I-
w I. I I A I _._11 11.11141
1 .JlIxIw I.I ._.. V F” V T.<_Io I ...I .A 11.1.0
r‘. I .rIv T YvoA . 4 I, 4. _ .1 H._ 01V “VI-JOAIO ..I-
»rvAv A TAA Al. _. .. 54 ., o _ .A ._ ..II. ouJI. III. 0. _.. _. . . V, V..»... _.I9;IOI
I 0 _II I. I'bnt _bf..fII,l.H*I.d Va II I. .I. . - ..1wl .3 _.QI'A
t . . ._ .. o. ._ .I.. ._ IA ..,I?Iv .I. ALI. I..I , ..IwIT ...I It TOIII.ITov v1
. A. . ~ _ _b 4 I.. I4 3 A ILL} 0.1.7.1. V A o o vAfi
u 7.06 .69: a I 4—» V.. .YI IOUIYIIrboIIIoIO. a... IA
1 .I . I III a .__. - IRVOIJIOUIOI... «qt-.IOAHn
. .I... I II. I in It. ._Ifi IF l1..,[0nu.._¢.voo-O
I 1 .. o 4 v « IIhI-IFIOItI.‘
__._. A — . ,wu. I.II...,.;.e »
.:a .. .,. I JOIT ,tvo-
. ._ ._+« .o ..v.#IIo.?I
..IvIV- ._I Iqu 0.0+. th0
f_ . ,H_« III. .,.A. vo_I
. it ,. . .Hw 3.0» .. 0L.I0II.4
I o ,. “:qu . con.+1v¢1 I
.I.... .. . ..:...r.-u..
.. _ . . .V+_.o. . Fund-41.11%.“ A .IA 1
.J... I’. v .. .r J_.
..I .+V. . I..... . V flMWHHATW¥ _..‘Iv I
, .V . . -..... .. .- .. . I..... .
I.. H. . ".00 . .Ahlw _JAIFIYHI._IMHfl+ A .A
.. . .a. 3.4.4.;
... v H. .I'LI Il.urAr.,.I4vfiaI AA
_. A .. 4 ..II __.I. A_ fi
e_ __ .. _. . ..I .....V .
. I. _c . 0. I0. I..I.IIIVIAI 1 ‘0 .A
v ...... “r. 9.. ..I1III.To.ocv .... o AvaI
1. V.- ..I. .I. I I
.I. . .:.... .I.. .....I0 .IOUOOIOOI
.o .I. .1 we. 1.01.1940. A A:
1...! .r._Io .. ... .. 160...... AwIA
1L {0.17.00 4.. HIT. .. .I. Ii. Qanaus.
II a. vol? 0. .v u . avoiOIFkow
II 1013IIAI¢I6¢IIII094¢_I J_..A_ .- I10. A. .0.-00...:
TAIvO‘o¢_.o_-.-ouoo. .__, o1+ L“ I.I..ILoIncI r I.I«Ia. 9
1150.. AT“ .__bIIoOono I..I-I ._.. hm ...... W? II.L.ooI-
114.1 UJJAI1 I.I, ..I ... V_r .. ODAIAr.'-OI IArh
A VAY . "our...“ VLIIiA. HY*..QMVA..v 1W 00 V. H
bVoDII. IA_II. .I,AI_. adv. D. II! ,T’. .V. A H AA.
YAITAYAVAYAvoAV..*....Hao t ... ..Iuevls ... ... .Ao- . o
0,. 0.01.. r d. . . ..I: .0. . h o A . _. o t A . _ I _I . I I _ _ . AA
IIIIAYV .A-on.._.,.IA_ . or. mnoofirALI ”JPHHMHIHT h .._ . ..I. fl I :va .A.A
Iolrt.~_1.ooc . w .1... +IIIOI0A 0.. VA _... t «.III. it. I A o I
o. _.r .- _.A . . . n .4 A ... V.. ..I? d .. - YAIAvATAIvvoA O
_ _ . voéILIOIo §I9IIYA . III. I .0 o .- .. A _ ,_ I _ _ .
0 IJINUA.II.0.<1HO coat . r. IlIVIIII.IIA_IIOO...1 £JJIIIIO.._I do I... I}.._I'wIAIII4. .Iv .wIc v. I II
VIOIOH...0«I.a. IF- . I_ 1..le 11.,1r IIrIAd._I 0... fi v Avb. Ono. AOL.-. . '95...- m | .I...t. .IA. .9 .. AVAIA
1 U.- +iH+I_4.IVp. ..II! .0.-I.....O. I.IIIIOIiIIv I DA 511 ..1. .I. II I.>. I-I.H9 ..A _ c . . A TAIL
o. o _.. p. . . I, .. . _ . IA 5L1 oHLIOI ... ”I“. H I .~ 1 I
IY A 0 ...V... IA +1.. “yr .1. I n.0,. It A OOAV 1 A A YAT-AIA
.. .I.- A .. .. ”.:.:. . ... .._. I I 111:». V.
o 4+...4 . cc ._ F . a I.....) .‘ .ATAvbIA 1A A g». I
O'LLI r * r&._.o_ O. unvdII udccw, > 1 O A M A 1“ I AI
.0. 9.... + II.- w 00. _ '00 o . w W 4 OAVIQIYA A Ar I” I » I A I
TAAY o¢_roo 0.. o-.. a o. .. .._ao » I6?! .. as. o. .4 V 4 OIAIA A11
..IDII’V o. o 0.000 n pl «'0. » 4 . 0v . A‘VIA v o
I.nAvdd6uo h __ _ 4 a .A A . A0. .A 4 Av 1
ur- a..* IflAOI.u.. I HI.“ W or I o 4 I A A ; a 44VAYArIHIO 1 H
4r&Ar6PYI FI .I. no I.. I I r . I|11 LIT 4H H
.0. 0‘0. ..|_I. I. ._ IAvI ‘- , -.V. 'II . _ .00.. 0 A
IIT1I0¢IO V. .H a .u ._ . .l . .9 r o .... .. .o oIIII. I I.IflI I I
Lou >$A4II _ -_ o. I. ._c _.4._ I..I... .. ..nvvaI Ito... fiYOIOIOIIOAvA A W A AArA
OVHAVLI . .0 Iohln — b III-.A IIIIJdIvI .1_ $.01 a . . . I I r" .._4 00 9.01 I 1“ v A A
.4 50“. —., o I '_- D .Iio fl: ....w. AfiI'III oak A.W00 A ooH¢AA %
pacvo IV _. 14'. III. ... II . _c~_ ..u. A A III .I.. .f I .0 0 AA A9AvAvaAv I
H. III 0 U V. I.. A <100i_o I.. . o I ..II II. o. _ o. - AA .AAv W
CITA.H.IH I.I“. an. _.o_ u.- .. .*o I fIfIIIIOOI IA .00 o. ._. _+rOI I..}
I. ...V . . A. ._vb...IHFIO~1,.OO . .r... .01.. . o 900 , .V A+IO.I..IIVIV AT AVA.J ‘Aoo 0AAA AIAT
op... _... . A. ¢ ..toA 0.“ +7 .I.... .I..VJu...III..O...4.I 1... uh I. .0. T10IanIIrIIuoIOAIH T1100 ._I 11
DHFF 694+IL1._—__ . IAaIrIlIlhifoli lfino_tu _co‘o-uoI'v HH . _a A... I. a ._. II...V.I. .__. co. oaflb‘l‘can... o IP’OWA; IAA HA0 4?»
o. .I.I....IJIOI r. o o... ..II-cl IIIOIOI+F...110c.oo VHAV ...w . .._.Iw.1 IJIIIIHIPIW I. q. ¢.r..« ..I - v.0Io I.» 4...... I... ArrAA 9“. .OH
+QIeIAé __fio .I -.I .IIQJaIA ... A. OV *I-fc IA.AJ.,. IrrII oI c .1 .._ _c u .o 0'. .II-iIHadF cc. . a“. .. .bo-I0$I§' Ion IrAYAIA AIvV.A VA AAAVAv
lIIIIIVHAfloo It AI..W..V.IrIo»lIc1.—.fi 0.. vaI _ ’fi'cI'oIo1 .1 .._ . ..I« c.. . | MAJ. o. t+.. ...:o voidinif IIOAI 7v. . fiA: AVAIIA
I|.I..A+¢.¢ VH- vIIIOIOIIII- ItIaQa _ o . 1 J. .:II‘VIIO. .nOP-L. II It “If 10:0. _ . .w n . III ".3. «1... o .1. o v I. . tI' _VOIV 0 VAv VATA A A A . Ah. A A. A.AIAA
I #0.. 14.1‘+i.0 .I.. “o“ ..:}..I‘a . ...... ._h+-w __OQOIl ..I. HIV‘ OHIIHW_ IIIou .rv bI.'.H. I1... ooh-W._o_ov ”I. IOQAITW .A. Av VAvAAA A
*0.....or o..L.IA tIIov. A '4‘...“ +Ich01-P__¢«¢o.oo ,.,.J.._v Iw.v. _.4I d. . n a A. on .H A” I..., uoI+ooaow A
90." A! o _ Iv». II} n . .... ._ .._. . . . . . .VI . ;IlII . ._ . ._ I 04.1....u _o. - VYAV
. .::... .... 1.7.4.9. ....HIHHII..-».:.:» . .. L... ._. .... . .._ -..».-.I V. . .-. .:.. ..:... II
D II V I I

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

DEPARTM EN T OF MATH EMATICS

 

  

 

  

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

        

   

 
      

  

 

       

 

 

 

 

 

    
  

 

 

 

   

 

 

 

 

 

  

 

 

 

 

 

   

 

       
  

 

   

 

             

 

 

 

 

 

         

       

   

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

MICHIGAN STATE COLLEGE

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

DEPARTM ENT OF MATHEMATlCI

 

 

 

 

 

 

 

 

 

 

 

 

 

 

     

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1 14 4 <<<1 414444 <<4J1d <4 4 . 44 1414 11 4‘4 4 4
oHaI-IHLO. coo“..._.. ,4«_m .,w 016 H.420 .I. I ..:-I..I 4. _ d I
l; ‘IIAII o I.IflxIOO.OIOIIII6 I. I4.II.. ..w. .I. . ..., ._ II. I I _ II o ., I I I.
l A Ioao¢ I o IOYrOOIO. OI0.I5.II , ..VoifiIIOOII .....4. I. . h I. I I . I I I. I .I.
IIVAVOOIOIIIOII I._r.o.aa 4.0414oor. .ovrap I 5910...? . ._ «V. II .f v II. , I I I I r-.
I o o o I . o u . I I I 4. U I o ”4. _- I I I I I. . IA II ] I. ..I. l OIII'IOIIIOC .1 1‘ ...V. . _ I I . I V III | c .I. I Ii I
111.090.90Ys... ,,.Iy .IIflI4oo._- II<Io..I.... IQQIIII4._. I I..I“I «IIII u..., .. .. . I,_I.II I .I , _.In, .I I _ .
Iv. 0.0.9. I .o I I....oo‘vc... oaooocuo... I........., I ..II. Oil. 1... .I...
IV. I... n. _ u .0 iglOIA,OIIOIP 0.... A W W.) .V.. I I I. I... I OIIIIJA 0 -..II.-
I I I..! 9| ‘1 1100104410- IIIILIII . . . v. .. . ., . . . . .I.!I.II,II 4 . _. Fl __ _ __ _ u, I. .A ..II .
I 0.7.5710. _n Mu ”04 >4<4IW1 «J .I . . . . . . . . . . ,. .I. ILITI.I..IJI IOII..IO.LOII. ..I. ..1 ,Itcudl..I. I. .II. ‘YII‘.’
IIrI. I.... 4 H HOOMIO. II . .. . . I... .I . .. I. . (I... I. . .I I. .. a; ._ _, _._ . II, II—IIIIrIAJJI. El 4
I .u. I «II4.II.0.¢IIOI._I I t I I1 . r‘.. . o. . 4. .I10IA
Uv . o I 0I¥. T4I¢_.f1. . _. I1 _0. .I . . .. o 4 I .. _._hL'-.-OCIdQ»
IVA-L P _ I1 I II... II I .ri +IW4 . I .I I I . A .E“ _ Q A I I _ . I” It ..lbI III! 0:
LII 0.6. .III i4§+rfit 14 0 4.0.49... . _ . 4... ow .4IIIOI II
III .9. . . .II .I I Y. .fl; 4 #1 I.I. _ I _ kl. A I t .4. I 0 c _ I I .0 I .I. '0 O.
I Y n. .I. r 6 *4. .IWHOIIIrtIA IJImI , .I.. Al!\. I. I Is I I . . .a I. _II .1. III
. WI o . y . . a. .. . .. I44 .. ..II... »I .I. ..V w . c . I I u . ., . _II
M _ .90 v I... .47. T _k . .. o? MI. 71.4.. 4.. . . H _ I, . u . . . .. 1.53.3.7
«I I I m. HIM >._ .IPIHI‘I .I...II:AIIIHIIHIIII.IJ.II. .llnwu .I . II_II.
1.17. 4.0 . ‘0. ,I I .thI 1 04 ff. .._,— I..I+III .I., . III III: . ... a I. I. ...IdII
Iv? to. IOoIrOJLVOA60#§OoA T347. 00. I..... .4. I I .I I..?!.. .70“.-
o .0 ..I. I.I] 0'11"... FOI.) I ‘v \r . It“ I.. , I t I __ ._a III-I ._ . I I
III . I I II 00.. .n v. v . o N I.. [..I I n I I. .. . .I II ...IF . I
Iv A ..I {I..IQIIOP Ir". IA EMIII: .Iu... . I .. 9A 4 _ ... .IIV. "TWIN”
._. I 0.7-?ttlo§ *IOLVIF I t P: I I. 1 IOI.‘ .I I0! .I.
.. O o I o O a I I .I I _.I .. no 0. 0 v l O Q .I. ,IIOIIO OI I.IIIOIII I “I ,I :‘ . __ > 4 n O A. O..I.I..|.. , II I. .,
Y. fll. IIIIIII... "ar,IIII+>I1 [6inon . I . I I ._.. n . m.) ., .. IIt..1n...¢I.. IOIOOOIO.
.I. 09. 60000 0“. #10 .uGJIIOIII lIoIr . I. 0 .I. ox ... I? I- I...I I. I, . .
Iquf‘ If 4 fl 1444144 flul§InF§uAl|WIIIIOIJHIWI‘III—I.J_ H. .. III..I.I II II... I +IIA._O SKI". Id”
II 'IOIOFI.IF‘_9L. .I- ....I 4. I o I r V.. 0 .III. I..I...I.fi II...I.I,,IIJ. . I.. . u. . . “I.I" ,II A... ..I _.r . .. ..II .ILUIalnIIIdI
I'lfi F. I I.I,OIFIV .0 fl . . I4 I. r .I.I.. J 4|. . . T1 .I .I . _II I Dr I I I . . , I . . . , . ,‘ .... . ,J I I ..I.I_..I+.‘II I! I 4 I
rIvO_.IIO\II,4._.OI, _.#¢O.¢.Ql _rro OI. . «O 1. III .. II 1.000III I u I.” _ ... .II.I.OIIOI+1.IQIY.46 .. I,
IIVO it " .v.4 .0 .I . ... .. 4,1Il.b$%n Io , .. 1.11.? I... . ._ r.» I. .. ,, .. .. I.. _. . .... . . . , . -VI .. :I
. H w . . .. I I.I...r .. _. I»! .V o _ _ . ,I II... .I . L II, , I . 4 __ . _ . I I I. {..II ._.L.r .
[II ..II It 04 H . . . . .. 1 .I....I. . .. I. I
O . ..III I. r+IWITI HH 4 ..IOIIU. 06.60- 09' ..II I... . 0,.0IOOIII III. I--I‘. ...
H1199 In; .4_Ir J .044!!! H k1, M. ._ _. O I. I_II" ..:. ._ .4.. I. on I.I I II III .
f . ....JI. b I a . _ ..II. .. . II

r I» I .. 4. .... ...I.... ..I. .._ _._... ....-. I.IIII. I..I . u 4. ..

RH - .I" hf l b? L . b I ID _ h ... A II 1 I V \ +4} .4....

{I AM «I <4 {1 4. <lw4w 4 41141 _.1 H .... 1 4|4J I.I.“

Iv I to ._ I ..I I. .I.. I . .I.. . _J_...vl..l I_II II. A , . I .J. IIII I
IIV _V Id 0 _ .0 #I .I rad I u r I .I I. 9 _II . . I I o . +9 I. II v I .
I . .na IFJIIWOI.1 I.. I I_. a. I... I... I .I. .I ..I . .II II 0. I _H
YAV. . , ..IO‘.‘O40.II <01. a109.‘ . ,tt. .. . . I. 4. .le 0...; O I o . O. +I II . I.I.r . I .I . ...00 II. * I I .I. .-
III a p .. , ... . .:I r . I I.I.lYOl‘ I..I.r CI... .IQ .Iv..l a In ,4 ~L§ .I.... I9. 4. . I u I 9 I I .. o 4. I I . .. I I. IwI I I I . I _ .III I I . — . . F
III . .. _ .. .. u.....m.. o. . .I . _. I “I. ....I. ..I_II.. .. I.. LT II I I.." . o
I , I -_ n o .I . . 9 .fl . . 4 I I AVIIJ..I I dlfi . II . . _II—I I I _ o I I , I I. . . , I. . . o l ' III. .I. .
nr . . U. I I .. I.. II II,III $6. I In I .. . o ? II I .. I . I I.. Q I I . I o I
YL +0 ,I I..» O «I..I...I ... I. _ .u .. a OII.OI..., V I . I I. .0. I _ .. I o . I I II I
_II-II I l0.v.l.l.§.9|lvw'IIO 4.4“ II... II ... rt 0 _II .I.! I..-IJII I. .L I I-I..IIOOIOB
7.01 no I I II." ..II. . . .I _. 4 . _. .. I ..I. .. I II, o _.. . IIII
LII .::. II I I. .U . F . _ I .0. n . I I I .. I I.
f. ..I. O I.I,IOII IQYI Q . I A I r I
.VI'n I... I... I.VII_II. r... I O I.
6 .I. I. I 010+ ‘0‘! 4...-... I ..I a I I I o I
I I .I. _ . I. I . I I , I I .II I
. .1 .. . ..:.T. . L I . Z
V.. I I All I. Y .VIIII I1 ._ .+.. L _ . r I J I. I I I 0 . _
v .. . .111 14“. ._w . I.I.IIQ tHA .0. V .V-I .
OIOII ' ..I. Ir‘lt 0.4 (tit-fl. [...V]..- .. I.. I<IIII 1 IIUI 1 .I HI. I FIG-.1 H
RV I . IIIII. (barf. . 1 I. .‘r :oIIIIII H1 4 « ._. I _ ._ ..I I .I . .Id 4 _o I ...< o
I" II +4.. ,J|.'.II_IOIOII_4* .__-OI? Ifivf¢ II... I I. IF. 1, _....rdtr.._._- a...
o u I. ._ If. [1-1 I .:H .. .... 4I I «I. 7 .II I _ I. . . ~31. I..I..IIIIII. .. _s. I
. , I.I.HIIIII r0.... IO..UI0. ‘ .I.. .4».., I_II O A. I.I..YTIII 1+qu.I.‘ o .I I _II J . .V.,Ict I;
51.. I. .I. . v .4.“- .._IIIIIIII 011 I.. a .0 wLI
o I14 «796-0 I III. a H 4 44 . 614905.... . . 4. I I
fiI I..II‘III. VIII 0 p 4 A a . 0H DIM! II. I ..u fl 1..
I I‘. 1"... I0 ”.0 “ AI; II . ,.. . . I | I. I .I I .o 1.1
T t 1 III In +<Io+ +1. I _ . . ...?HIIPJI r4 ..
4 . Ir .0 of . o .4--_II I1 0.4... I... <J+IIDI 1.... __.OIIDIIL .. m
Y1 .mI...oII ..II... 0.... LI .4. ..4. ”I...“ I ...IW!HMFII H. I! III.4.I.. v0 IIwIdIo..H.4.I .I.. I..IIIIOA aovoto, 2.. 4
.F 76.94.09I4inf0 u. I I... _..440 IF. . I. I IIIITIITI, 4v-r OI ..r..._..._I%...w1I._FOIQ..1¢ IIIIvIIIIr.IO.I.>
4o¢oooadI4II64II .o v+oo++._f.o+o§&+Frr_I.+fifo.¢;g.4..FI1.11.56 44.44.. ....IIYIBIIIIOIIIOIIrYI. I..IIII
.4f401I4IIHQflIII..1qJQI I..I ".IOOIITwlan. In +II._th. 14111414....w‘. ItgLIlv _ I. .- (+9 .I....AIIOO. PV.O.OAIII OOIIIIJI I
v ._ Ifiwbiv I..... .I. 0.. .rd 0: Iv. 4I4I..4..._ ,.I— .94_I4IIIMI,T..I.th.ol-I.wl I I. . . o L—I..I. 1 _II .IQII... ... OIIIIOIIII! I.-I!
I. . ..IIIIIIIOII.III. IIUIIwi I4I F¢¢+IP1I..Qr..._a+0'.I+HIoo¢r> mm I 4;...I4I. I¢.Io¢.¢+ IIoIIoiII..v6I¢IIAA III
. .v H I . ...IIII..4III,I' O I. I IITHF .l : r . _ I .. I .. . o H H . _ . I o I _. A
JIW I. 1.I.I_.f...,. ..I 5...... ._..” .._ . «ML .. . ... . m n
. I N II, ILL H .I. _I
a .51.... . «._J... I. n ._
.9. I (H +‘Fc . I ._ II 0...») ..II.I+IIIII U A
II. . I. _... I ..I tut..!00- . .ooa. no III. I.
rIII+IIOII IIooI.I I I I FF 0‘ ..IIIAIQIIII I. I . _ . IIIYIOII. -
Yf‘cdocI-Io.o¢.o. i _I I II rIIOAIo4¢I4QOII II 4..... ;.. I I 4.
vaao ..I yIpIOIOI ,I o _ I. II... .?.oor L... ..4 .IInw I ..... 1
17¢... .. I.. .. . I. u. .q 0414744... I. _. .IIO...V 4 II .. l
IVAv¢.o 1. 00000 I. taut. ..I'OIOJVIO r0 Y—ODIIOF I.. .W I
IIVI+II IIMOI .II . I.. I I. II . .. IIOII§+LI<I516I0II .1 .I.. I.H .4 ....I ; .
I III- C .. 1 I I. ..I liuurtlfgtl I‘ .I 4 II.II. . ..,. I. J
T. o 9.0.00‘0 {OII h , I .- I..I...04IIA a H «I... . 11.. .4 H I
TIL on. .- . I. I 4 I O. I.».. I n I. 9 ' I _II ... . __._.. v0. . 4 ...,L. ._. .p . I. III
1 II. ..III.- I I... IIIOOIIPIP9I00I II - ...“ .:I.b.vva.I. h... .I.. 4 op‘DII IOIIOIIIOLQI
I I IIIIOOFOII . I II. o 0000.... ".qOIQOI-L I I IIICI .. \ ... _. ... .. Iv! II II. Hm I ....4 ".“u. a I QIOIOIIIOIIII4JIOI0
IVY. .. * I ..II. III I II. 0 IOIOQOQOICIIFI ... .... 0.0...4. OO'OF4.OQIOIOIOQ O A A4IAO.‘ .OIOIOIH. DQ’OFIJOQIII.¢LII HA
I: 4 Icon How-or I»? o .I. I ..I.._OY§.IO .0 II. .III ... I , . n H. . «_.uNIIIIII I... __.I. .::. I.I. ..I..AIIH. .414.
I. ..I. n».......vo¢ .I.. . e... ... . ._I_..IIIIE..t.. III. .3 . u. v I¢_.III .. _. _ 4.0 4......
..I......t.onto‘III.II II. .V.... _o I.Iooooo LLIIIYoca_«-.._ I .. III ._I4I_. IIQI4 . n~_- I.t III-.II_I.I..,64I ..:4; _. r l L
Ilvxoouoon4cnao .. ..I ., .n ”...... .v .Iv'llwoa_4oo,_«. . _._4. I0 thF+oIoa . 44... . . .94....164FAI4... —.

4 I Dl. I. .II _vllr IIII' F.. F Ill. r 0 I ”hr ..I-.II .I.. I..... .I. _IIII ITIO IIIH ... . Jilli— I 4 . m . .I.-a I Iii-.I.. {IO-J “III «Tp _. 1 I
II.II,III4~ 0 .4 ..I. .— .... .__1 ... ..IIII.I o .,I....,..II....I_..__.. . __JoIIIII. .I...I.I..,, .nw _._ .. 4..... HIwHIIM A «IN
III,II._VI ...4 cont v .7».IL.IQLII!I.1I .... , vfoooaoohv._ooIb.QIFQIOIII.._..IIIIQI .4... I +41F 4I.HI., u..voII TITYIA YLI
II'O,OII.O...I¢AIO ..I .I. II .. s c I O ficJ’IIQIOIII . . I . . III I..I II LIQI. I. _. o. I ”I... . .III,L. _II—(IIIII. I I..! 'IIIIIVI I01V0.*A
ILYO t I. ..I... _ i Q I d To. I _ : . ._ . . , . ,. _.. , I... .r .. . c _. III ..ICII I _. I I ._.... I thin, .V ..IQIII I YQAvo‘AAI
IXVOI’OOOII .V. I L I II I I I‘I‘OIIQIOIQ... , . I II II I . I I I .0 I.I-I. . II..._-I O . IIIII u: O'IIIII 'I IA. 71

4'0! I..... I I III III -! I.IQIIIr ..I. III... I . IOIIOIFJIQIVQIAI4'III.IIO III .09 LIHQQIVQOOonggLJ
WIHO I IIII I a I _o ..I I . I. . I _ . I I I. I III I¢04II I.I.AYFLLYAT.
FAWN I. ’I 0 III I _. I . I _ .- I..I... .Q rIo I..I...II OI.4
I ..IIII4 ..I ..I. ..ova I . I I.I I4. I..I. .0 v0.01
40 I «I III 0 [I..-.40.}! AIIIJllbO I. I.I 4 1 I 44 0
I . IICIr|\_. II . . ._ I I I OIOIAI'I.L.,‘.I I II 9 III IDI. ..I.
I I. .I¢ . o . I , I ..I .IIII.|.. III II 9‘ o .. IVIVI VIIii . ......I ,v. a o 1 hp; IT
.. 900.04. ... . o IIIOOLIIIIIVOI .KIIIIIOIIIIIOQO vI IOIIIIII.I.I lrr.
fl; .0. . II ... I 1.... ._.IIv. ..,III I ,.|OI a III-441.0,:I‘AI1
V II. I..II. I . . _o o .I .I .. ...i I. ._.I004o4 IInI
I , In. ..I. I . . ... .... a..fl.¢ ,. III. I I .. .I. ..I I. o‘V-éi
v1 0 I. I , . .. . ..o.. I 4 . I Avid 00.01.... I.I.
I I . o .. _ . I . , _4A o . . . .I II..II I.IrIQ II. I.. L
I . a. . .,. _ . .. . .4 . ...I. I I.. ”I”... I
«O F .. . I Q OJ 4. .1 LOIOI‘VI“... j ' .9. . Qua” I rAI
. o . 0 I _ _ I I I I I I I
I a ..II. I . . I . . . .C Iv I I. v. 9H
1 I. ._ .I II I I I o. I I II I... I'IIIIIIIJVI_¢M
.I Z .I . u I a. a. I IIOI II I I. u. .I O .flI IOOOIQI 1IIA1JV I.
U I.I. ..I. no. I 1 I I.. .I I ._ . ..I... II TI 110360. oorlII IA
U .I.... c I I. I .. I.. I I I I I I .09 $0.1..IIItId II ITAVAAJ
..II. 9.1 I 94. o .l I.‘ o . I b I . . .I..- A .I“.
I II I .. I .III.. .... . . . I .. I I . . II I I 9. . . I I I I o o I I o I ..II I.I. III9.I.I.|II.J o
IIIlY-‘+III,,I Itaiui h. ._IvrIVIIIILI... I! . II I..: I I II.» . II. Ill—..II . IIOJ ..II: .I.... .I.! h I W 4}?
v .I. ”a... “..I (o...~.¢ .. _ IF I . In . . I. .. 4.. Y69I60l, IIIIPI CEII. 0.0910. 1
v... II D _I I I . _r o , ..I , o. . . .. I I I I I I 4 “ 9..Ii .‘LIOI'-\ OIII I'¢ O 0.4.”an
II I . IF no. «IIIN .. I . I I I I. I III I u .I.... I 1.404I I..I. I.-I.‘err
...... I.I..III_ I.I-... V:- C... I I I ..II.» ._I I ... I.. I“. u. 40v.‘ III.
..I. _I ..:..I I V. .I.. I o .0... I .I.. II _ I ...-.V.. 4.0.0. It. Y'VOALIII
7H. .I....i: _ ‘I 504.... . I. I III 0 I I ..IFa , . . ...Frc. O‘IIOIOI.IO 0'+ H
LIIIA ... II.00.49 III... I. . I.. . ._ I .....F 9?... 4d IIIII no... oofimWH
v4 . HIIFII. TIA I III 4 u. . ... 4. I ..I . ....HIIIIIU OI. IIOIIIIIII cow:
IIIMHVIMG it... .2 I.. Tr... _ I. I _ "V..... I-” u. “I.I...HIIIJH :3:

 

 

 

 

 

 

 

 

 

 

1 ‘ ‘ wImHIo

 

 

I—I-Ln—IL—J

 

 

 

 

 

 

 

 

 

. m a} oummao

. . P L
W 7 #028 _
M — . . $3-0 a .
. . V ‘ ..
a .

Stems

o.mfl‘o . ...__ W., . ».¢

 

-.. v Y~""W'

MICHIGAN STATE COLLEGE
Daily‘Ayerages

 

fi 33 U

 

 

 

 

 

 

W m m-mfl-o _

W m — ocmfluo L

w a

w .m fi‘ xwunu _ _

 

mm m m . m s s u .2 a

. .: movduchnophmo m

 

< ‘ . . . . . .‘ . .
Fit‘kD" .. . \ I. I.‘ Illll v ... r..’\'lbi 0y- \»6-D‘.l||¥..ll-‘rlllh III: (III. A

MLNY OF MAYHEMATICS

“AF—{7

LIL

MICHIGAN STATE COLLEGE

--.-o-s-‘—— xvii

—f."l.. _-
‘ . |

O
0..

_-Tr

.IflI‘. . .I.. ’LJIIT.[

_—

1. NJ..-" vhf -_

___Lw I Fl turf...

ooo—vw-Ov—vq

5069'..-

troé-v—O-o
.050 00-.
.-.

l4

W

I]k

a

..u. Ln+ct _-
_
1..L-

10. 0% ,.II

F

I.I;

_

o .
o _
1 . r;

.... vlu

_
..
_

.. .. .Hu-Hf .1. El
I .. . O Il_4 “.._v. . ..|:
t

0-0. 0
.J_
_
.370:

llL. ll .Lpiltrlllr'.

_fio>vv1 . to My
_.VIVIO.O..l.I._t.C-_..r_I”
9009‘ {...V 70ak=
.I..ID|‘.... $0.01..
(..10.

 

b

00.04"...

 

 

 

DEPARTMENT OF MATHEM -

“'3“ '5‘ R00... use ONLY.

 

 

I
I
l
I
l
I
I
'
l
'