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A STUDY OF THE CORRESPONDENCE
BETWEEN POINTS OF A SPACE OF THREE

DIMENSIONS AND LINES OF A SPACE OF

FIVE DIMENSIONS
THESIS FUR THE DEGREE 0F II. AT

Margaret Grace WaIcott
193?.

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remove this checkout from
your record. FINES wiII
be charged if book is
returned after the date
stamped below.

 

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103.5139“?

“T"UY or Txr co“D““Uc“r""cT .tT*TT1 POINTS

Didi. i is;5._1-.1. c... ‘44;

H r '1 1-1 mTVfifi—a T‘T .w—xrhq’fi ‘Y'N '-.."' T‘Xfifi
OF A SP ‘40 ‘ CL L:-¢L;J._J 1-1;. - 4-:VJ-t‘v -no A- D Li-IU Ho

CF A 3? ”LC"... OF FIVE DILEITS GETS

L4

I" :ooUo TI

It is the pu1pose of this cape to discuss the correspond-
,ence betwse11 lines of a Space 81 of three dimensions, and

oiz ts of a certain hygierouadric Q2 in a flat Space 85 of
five dinen ions. he dualistic transformation between points
and planes of the sa‘. 2e 83 is well known. However, it is im-
possible to make points and lines of the same 33 correspond
unless they lie also in the same plane 82.

The first part of this paper is devoted to a discussion

of the general correspondence between lines of an 83 and points
of an 85. The second part is concerned with the correspondence
between certain line loci in S3 associated with the tre ns-
formation’ C of nets and point loci in 35.

Let there be given two points Y and:;, in 83 with homogeneous

projective coordinates (14,, £2 .V3Jfi4) and (1,41,73ng ) resyectively.

The homogeneous Plucker coordinates of tne line 1 j;ining t to 7

may be denoted by ‘gn‘g% 2% lfizéljfl where

, V/I' ,, ,
j, : V, (111’ - fig 3, 7 j; : fl, gzwtay/ 2 ,03: 13215..
(1) /

It] = f4 ga' Q:— g? J I
fi'V.G. Grove, Transformations of fists, Transactions of the

merican lathematical Society, Vol. 30 (1323), o. 483.

:D

'7 rooL- ‘9‘; "
'ereaiter reizs ried to as Grove, Hets-

These six P ucker coordinates satisfy

1“ 2 .. .—

(4) Q9 " 1:1!" 12.15 Fj3/y-

identically for all values of B7 and gz, i = l, 2, 3, 4.

cred numbers (Lita, c, is £154.)

"\
kn

We shall interpret the six or-
as tne six homo;eneous projective coordinates of a point L
5. From (2) the point lies on the hyjerquadric Qi.
We shall say that the ooint L is the image in_§f g: ne lineiz
that I is the 137.8258 _;"_ L. If follows

A

~ 0 a o I; o
tnat every line in 83 has an image on Q4 in S5 and every

in §_ and converselv

C

c . 8 . . .
p01nt on Q4 in 8 use an image in 83. These statetents may
be summarized in the following

Principle of Imaveriz From 8L: statement or theorem concerning

 

 

 

t 9 relative DpfltlUR f the lines of‘g geometrical con1iguration

 

 

 

: :.: .- ,..,4..: , 1 ~- «7 : .
in orcinary prOJective :nace o tnrec OlmsJSiOlS

~ -: an“ «A -14. ‘- - - —' 4-K .1.-
ooteined another that:JCLC r tneorei concerning use lglfblve

 

fosit‘o

'1.)
:5

 

‘. a 3.: l. '_ ' - .Z‘ ". «an
experquacr c *3 ? PIQlfiCtlve §33ce Of il”3 OlYDnSijgoo
_—

 

 

 

 

‘

an 8:5 as forming a local t5tffih;hr3fi of rsrerenceo Let
coordinates of these poin l

hedron of reference be 2;“4;,¢;

) a

We may prove that the coordinates of any point 2 in this

83 may be eXpressed in terms of the coordinates of fl./k’¢,7'

"1 . ~“.‘ A “ A f"
“cn1s1111r "the 11qu1't12u1s

(3) .Z =t,f1'+t1/££+ta"é ”‘47": 1 = 1, 2, a, 4.

CI)

The four ecu tions 3 mar be solved for the four 111-11.;1‘ Us
,

t3, t}, t, ,fv in terns of Za,/€z,/4u',gu'- The as so itions

will be unique, since the four points, 141, A», g are non-

coplanar and therefore the determinant In; A” 1¢1 %/ 3561

It is a well known fact that t,,t},ta ,tq are tile coordinates

of z with rse ‘oect to the local tetrahedron of reference 4*, 0,3:
we shall denote the lines joining flto? , é’to/L, flto 4,, g tO/t,

L3 to» , AtOA. by FIE:72 E‘NZ’L respectively. The points

fhg p g 7? -K the 1 ages in 35 of these lines in 33’ form a

non degenerate pyramid of reference- The coordinates of any

point F’in 35 may be written in the form

(4) P= r.€+£1§+r,~7+nf+e’i+"at=

:1

q
I

:;e1e 1., f1 , #3, F9, is , 2‘ are coordinates o." P with the

py mid.ޤq{f,{as local pyrari d o

1-},

.D
17:: 1835331

0
(D
O

l. Ima bes in S

(H

of the Lines of a Linear Cong‘uence in See

The points of ’0 joined. by lines to the points offl in

every poss1ole W3“ form a linear congruence in 83. We shall

5 in 35 which cories ‘ ni to t1-ese lines in 83-
We may define the points R, on f and a onJlby the following
ezzficss ions

:: K‘I’IILfi,

P.
P,=/£t(‘5'

The Plucker coordinates of the line 'oinins to P will be
O I 1

of the farm

P =(P,,Pa) =(€+A«j, A +64")

=0KA+H&)+A(%,A+H4J

(-‘x
C:
\J

= (16AM Wm)” (WM 1 a (we

= “E +H7+Xf+lél77.

4.1.

The local coordinates of tnc point P defined by (5) are

0
Eliminating A and f(from the above equations, we find tne
(s) 14=0 Vc-TO fi’v:V1fl-‘"

7

(8) are the equations of two

*3
U
(D
Pb
H.
H
C I
Ci'
c:-
O
O
I" i)
(D
k )
3:
{U
c+
I”
C
:3
U)

‘ 0

F1

' u
,4
(’1
'40
(~,
:55
"J
O

Lyperplanes, each determininx a Space of four

K.)

Tnese nypernlanes intersect in a Space of one less or three

state the theorem;

 

 

 

a - fl - F - ..- .2 1,1. 11' : a .c- n, .' - ..
Tne M inuc in 85 COTrOSJQanhr to the lines 01 a linear
con;r::nce in “q lie on two hynernlanes and an ordinary

U "'"‘_ —" —-—-—

 

 

an 53 in those Qgperplenes-

2. Images in S” of the Lines of a Special Linear Complex in 83-

Consider all 0 he lines in 33 intersectingF,. These
lines form a special linear couplex- Any point on the line

is defined by an expression of the forn1z+fig; any point in the

5‘

plane determined by % /£/¢ is defined by n xpressior of th
) 2

J-A

form 13+ u,¢+-fl4,. The Plucker coordinates of tee line joining

these two points (any line intersecting'p ) are therefoge

defined by
P =(Yt/‘j’ g+yi+7jfld

n

w, %)+(l(f£,/b) + New)” H(g,/b)+r(19(‘3lfl-)
= p+¢4f+oz7+1éff +105:

-, -~ 4.x,, : r n +* :3, a A-" . -
.tes or tne point P in a , tie 1184c o1 p 1n

7%
a
H
O
O
Q
.,_J
O
(

O
H
5.. 3
F-
:5
D

5

(J)

16:, fl‘tfl) £337): fiVJ/(H, £53/(fl’fl6:0.
Elixinating A)({,z) we find that
(7) 74‘ 3 0 J ‘Za fly: ‘Zat f:-
The first of equations (7) is the equation of a hyo;r lane.
The second is one equation of a hypcrqucgric. To-

Y‘ v ' '1”. .v V "1 " " ‘.“. ‘ 'L" "V F - .
regreseit a cone hltd we teh at F’(1,O,O,C:C,O} ena veb CUaerC

 

1
L
I

Cf
rd
H
l-’
L)
(D
U)
0
F!)
(I)
"1
O
l,l
“‘3
r...’
f, H
i.)-
‘1
D
1.)
*‘J
r)
L)
rd
(’)
M
1.
’-
(D 2(1)
:5!

 

 

with a point for its verte: and an ordinary guadric in an g3

f r its base.
3. Images in Sr of the Generators of a Gone in 35.

Consider the lines 'oininrr to the noints on the conic
E3 ‘3 1
1.. _, 4' , _»*. +1-._ fir: . * ., -.
fl, — {(1423 7 24-0 re..errcu to UV. local tetianedion 1/, ,{ £6 a
We may write the parenetric equations of this conic in the

form

¢,:f(/(, yzrf @zX’, fly:0-

Tnus tne conic fl”=f( {4113 ‘L/‘fio is.
Z defined by the xpressian

Z: ‘HAZif/t flap,

8

'l
. . w, n r; 4., s ". ‘ ' ' .‘ - _ ,
T‘s Plenfl' coorulnates o1 tie llLES p JOlnlng a toz:mrc

V
II

(L3)1)=((3,f(/(f+fx/A +1342)

= "37((£;3)+~k(%144 +x=(%,rd

—

=-—f<,kf+1t?+ 12);,

4.'-. ; ~-

The local coordinates of the point P, the image of one ine p,

‘ .33-,
.re tierel re

7‘.=—K’A, far-o ¢,=o,g,=fi’, {fr—F, 126:0.

Q)

Eliminating,4 we find that
(c

U) 142,30 flgzoaflézo flagfi/f‘lts.

 

 

es in Sr of the Lines of a Ruled Surface in 83.

H: uni-+0.31. +a5p+0s¥¢d’
Be b,w’£+ba/£ +‘b3A, + 19,3;

\nerein «4, M are functions of t each describe a curve in

(I)
()1

. The lines p joining points of A to points of B generate
a ruled surface. We stall find the i age or this rsled surface

A
-~-

in 85- The Flicker coordinates o; p ar

('1)
/\
P’
(D
H)
P.
{:5
(J
c.
C
s:

“J

P: (any *m1A+agfl/+w¢g, 6, «+6a/L7‘bafll 1‘5ij

‘€&.bq—Mb.)<14, 3 ) + (an bra/16,) (1!,Az)+(da,ba-azab,) (164,)
f ( 0w 62.- Mbq)(3,»)+ (M63- figmflgWHM/sba - ween/0,41

I (0..|bq-a.., 13,)? +(av.b..— ax..b,) 'g’ *(a’lbs”a’ab, ) )7

Home“ on I»)? + (“v4 be“ @3va ’7 * (“’6‘ ”‘ 1,31,)».

Hence the local coordinates of the irage of y in 3 are

 

 

 

 

 

 

'. 1 n . ~-\a :' C. j- -. , .9 .. ‘
Tne poizts in or CGEIBLLQKVlflgrtO tne benerators 1 a rdlcc
—— —‘J ‘ -— _ —
f‘.
- .9 , '., n ' 11‘ 1.11 .. . a : £3
suriece 11 :7 lie on a_curve on tee Lf‘eZApacr1c Q 4.

s a special case consi‘er the i ewes of

.r.‘ n i .- 11 . ' . ,,- .m-
01 a re ulus. Tne point fOnS dei-

-1
:5
(D
Q;
(5‘

1‘ a
1
c
*7’
b
(I)
g)
f I
,5
’-
L')
l

on F and /£ +/(@ On £ are projectively related. The lilies

h
(I

( fl'*'4 3, ,4 + A AA)

W,4)+A(z,¢) + /((<3,/L) +1211};le

Eliminating A we find thvt

 

 

 

 

 

 

 

M“ - .L‘ 4.11
-“c1-131c, e Hive b-3 L ears :
"" J— R “ x n 1
T 3 ;»+‘b8 » 3p C rr s o L to b-3 3 e19t1"s 31 a .\ 11‘s
-— - —_ —-. —
Q
q ~‘ " fi-V -~ 4" v r - ‘ «
.L....(3 71.- C ‘ OJ.LL....-\ .1. "tr krl .. .L . L 4.1 :e .1 - j: 1! LS °
v" H .2 ,~ 4.- f‘l
c. I apes in o- 31 toe Tan - Us to e Co 10 in on.
v‘

‘ O ‘ U _ ~
Tne conic, mnose equations are

1 1
(-0) ya, = N a :23 , 21,:0,
lies in the plane of V,,£,,¢. The parametric equations of this

I
COXlC are

nngent line to tie conic has the coordinates

The exnression defining the line p, tangent to the conic whose

equations are given by (10), is, therefore,
P (KAnyAfA‘AJ'fi’flf-lxmj

.. K3; +-K,("y 441K141.
The image of the line p has the local coordinates

N

y,=0, lgz—kzy £3: VA; {4:0} 23:0:f4=—2K//(0

)

Eliminating A we find that

(11) ¢,=o fiq=o,t=0 4mg

’ 5' 9

Equations (11) represent a conic l in; in one of the plane faces

— fr EC);

of the pyranid of reference. It passes through two of the
vertices, the intersection of fig=o , 1'==o and the intersection

6

Of 1%: 0 : 156:0 . It is also tangent to the edge Y‘s-o. we
may state the tleorem:

‘a

 

 

The locus of the points in S- corresoondin: to the tangents t

-0

A

conic i s6 is a ordinary conic lying in three hyperolares.

8- Condition Necessary for a Point P in 85 to have an Image in S

1-!
U ‘

Consider an? oint P in Sr defined b" the exoression
. o J

(1:) - P:t,(2+t,€+t3n+tyf+tsi+zfia
Consider also any line p in 83 defined hi the expression
(13) P=(16+/(«a+,aA/ch+,(,¢j+(4,A/)

= ”FUNK!“ 1",}? wuyff—Mmf +aa,%.

The image in 85 of the line p in 82 defined by (13) is therefore

(14) flaw/mm!F+Hfl7-x,yiua,iweé.

Therefore the condition that p in 33 be the image of P in 85

100

is

(15) t' "A “:A ta fl, is "an 3‘4=’/‘,/(2 fS:X«1Jt‘=fl//l'
Elinlnatinb A,,(') 1”: g, from (15) we find

t.t‘ — t 1, + tat4 = 0,
We may, therefore, state tie theorem:

A_noint in §5 will have an image in S if and only if it lies on

 

2
the hyperquadric Q in S

 

. 2 -
The quadric Q4 in 81 has two oi ‘uict types of plane

 

first kind if it is determined by the images of three concurrent

 

 

but not coplanar lines 11 §_. we shall say that a plane generator

 

is of the second kind if it is deter ined by the i 0.308 of three

 

collanar but not concurrent lines 'n §6'

 

Suppose that the point P is in a plane generator of the
first kind, say in the plane determined bylp, {,27- The point
P is defined by an expression of the form
(16) P=t,f+t,f+t377.

It is obvious that the ina5e of P in 83 is the line p defined by
=(1c,z‘,43+t,/L+t3 A1,).
If the coefficients t; are functions ofz‘, the locus of P in S?

0
defined by (16) is a curve lying on a plane generator of the
hyperquadric. The locus of the image of P is a cone since it
consists of the lines joining y to the points of the curve
traced by the point 2 where z is defined by the expression

Z 7‘ t,%+t2fl+t3fl/.
Knowing the type of curve generated by P we can describe the
cone which is the image in 35 of the curve in 85.

Suppose that the Toint P is in a plane generator of the

second kind, say in the plane deterrined b3 t1 e points g’47 16.
J )

11.

The point P is defined by an expression of the form
(17) P: t1£+t3v+t‘¢_
Consider also the line p in 83 defined by the expression
P=(#+AAU y+HAJ
= - A {4'fi(.7 +14514L.
If p is the image in 85 of the point P delined by (17), we
must have
uA=dszj H-‘Ats, Afl-‘dt‘

9"‘q
' I

”herein 4 is a factor of proportionality. We find readily that

Therefore
A:-§;‘ , H:’%“
The line p joining the points defined by
t3£+tbxi7 t,)'(*f‘xz,
is the image of the point P defined by (17). We may state our
results in the following theorems:

The image in §3 of §_point curve 15a plane generator 0 1e

 

 

.D
.L

irst kind of the hynerouadric islg cone with vertex at the

 

 

 

paint of int rsection f
ing_that plane generator.

‘L

The image in §_ of a,point curve in a plane generator of tne

 

 

ieteriining that plane generator. We may, therefore, consider

 

 

 

the image of the locus of P in sucn a_plane :enerator r3 he

envelone of this one parameter family 0 lines.

12.

7- Relation between the Local Coor 5 inates of Points in S and 35,

0]

Let the local coordinates of no :ts of 83 be, with reapect
to the local tetrahedron of refeience,j¢,1l,fl,,go Tfius re may

find the local coor di~1wete of a line in S that is the Plficher

:5)
coordinates refe11 ed to {a Hg , /Q,,LJ . Usi: 1g; the pyra...idf , g ,7
i, 5 ,1, as pyra:zid of reference in 35, we may set no a local
coercinate sycteu for Joints in Sr. We shall find the relation
between the local coord ' ates in both syete 8.

Consider the line in 83 joining the points 2’ and g'cefined

by the expressions

% 2" fli‘f £11 + £3 /€/ fig! 7)
%= L$~€+—% ,£ r %3,4, + 34 a
The point in S which is tie ins e of this line may be exrres sed

as follows
P:(r,z+t,,£+£3,b+fl.,lj,f.1’+ffa/‘+ga¢+§lv 3)

:(tl‘a‘l-t'l ‘3.)F+(fi,1;-fl;3,)f+(1,13“£33')k
Hx‘ffle‘tefiJf +(zyya-tagy)}7+(tagavx/sgd
Therefore, we find that
t1: 1‘: ‘fiqflfiq‘j' a ’1: 16723213!) t3: 2'33-‘fl3‘3"

magi-raga ts= Mr :34: Vega-aya-

Let
1": 14(0. 1)), (J‘s-(3.10.11), i=1, .-., 3, 4.
be the parametric eountions of the surfaces 3, and 33 . Consider

‘3

the parametric nets N, and F on St and 33 as being in relation
C. The functions fig and #; satisfy a syst

_ . A. 3 ~._ A _.
““ticLs O; tre Iorn

in.» Atw+6tv+P¢+L%)

'2.“ mtw+s¢v+w+wa, mgr/mm,

(E3) E,,:y£‘+lt,+zfli—Ntfi,
‘3“, :mflw'f/lf'lffl,
\Av =Mfivrgf+8%.

Tie coefficients of (13) satisfy the following integrability

Conditions’
(1.“, —.{v +0.5 ~ Y/j +C = -/W:M,
bv“5u.fb¢‘fi7 +c. =—mr‘\,

bw‘fiv +a¢+(b—a)b ~5p-P 2' 41L,

Gav-7..+bY+(¢—J)a.~74—3 =rmN

QW-Pv+o.Pf(b-A)C-Z(5 = gL—{NL
(12.) c _ . ~~ m
v zu+bg+(o.~5)Q-PY { j:
' Mw—LVi-aLI-(b—AHA'NF : 5L -qm,
va-Nw+bN+%amF)M-YL ==HNvem,
wv*‘r’m-M)q. "3 3MB, ‘
w+-(4o- )b ~r
m 4,, X -44”,

u
u
.n
I
vs
31:

3.» - {v *Law-lwa)c
I3

0.» ”'31 f” (44,- «m7M

H
.0

If tte fourth and fiftl equati ns of system (13) are iiierentiat-
ed rith respect to u and v and the values of‘y ,
My wv

tle first three equations of (18) are substituted we obtain the

following systen

’Grove, Nets, p. 434.

N-
E
u
as
s
+
‘“\
a;
+
In
Ԥ
\.

I=A+£+Tfir4, {Jean-P, F‘ML+W(’%)w-fl‘('fl£-m’i§#’

t =-~n[-<-‘+fi,%, +£.-P-(,£)w],

-. bf'ah'w E=flnM+m(%¢)v-fi.i‘6f1_@;fi_é)

M =-M[wn‘4~: +523*£3;. ‘Q’béfl, 3,

7.4" f= J+-2+:’f,"'+132 {=m~+m(£),-N-fi%-%’,

i7: ~n~[)’&;+73£p*W&% ‘g"Gth;7

The first derivatives off>,.L et:.
linear co:binations*’of{>,.k etc.
f: (14(5),
ue find that
Pw= (T‘wflj) + (It, 3/»)
= (xi-£12,?)H-{4/m/Lffig)
'3 (4, (a) + (fl-fiflagpm (14/1,)
'=(H‘£Jp +maf'ff

 

 

* v: * . - .2-.. -7\' no r . _ r N .4- .
L.p. Lane, Tue Piolectr»: ulI-€L~ tldlgfiGQ eti¢ or ; cbegs o
—. ‘oz- #1.. We: a 4-H u-A WW 3:” II— 5-

Linear Feroreneoue D$J‘CILTbLCI 2o etione i tre -iie Cicer,

 

Transactions of the American In

735.

w

In e si ti ar manner we nan find the first derivatives of f ,‘7,
etc. he explicit e: :Iessi ns for there are
P“ = (H‘£)€+m€-{,
Tw=Jgf+p7y+Ltg
7)..=(b—£;)b+(o’+1:)€+h’\(2+fi,
(2’1) 1,:(1- %)f+ 4335—5-13?
M )
14;; §*)b+m(oi+§)§'—mp\p+:éa «A,
A“: (4+£~+b)L-%f-M?+£ 7+Lfi,
9.: (B-%)Pt’”’?"7h
fly: InrY‘gi-Nf’
§,=(s-g)r+<bré)v+M{J—4,
fiv=(i’gv)fi+%??~zfip)
§.=(3.—§ §)f+ +§<E+£)‘—Jn/TA€—'é,d,
4». ~(zr,.9.m)1~-§z-Mfi+§?+~5
(21) v.'e find that
Pw-(H-%)(v= f‘g—_
Therefore the tangent. to the curves v = const. on Sp intersect
the line joining the points g and in the covariant oint
mag-g
Since the lines ; and f intersect in Sq, the line joir ing the
u
POifltS f and i'in S5 lies on th: hyperquadric Q2. To the co—
variant point in 3 must corresnond a covariant line in S3 with
the coordinates
mf—f=m(fi,/L}-(€d,/£)
:.- (m‘t‘8,4)
Thus the line joins the pointg¢ to the pOint/qu_gr, that is to

_L(,o
one of the focal points‘ off .
By symmetry we fird that the tangent to tip: curve u = const-

cn SP intersects the line mi? in 35 in the covariant poirt

xn b-$,

This noint corresoonds to the covariant line in S joining the

3
point 4,to the other focal point off . Te may, therefore,
state the theorem

The tangent line 0 tne parametric curve 1': const. (2.: const.)

 

 

 

on s at intersects the line {g (b5) i:: a. noint. Tne ira'r

)
(D

 

in §_ of this noint is the line joining A.(A/) t. the :ccal

 

 

. ., . at:
of the second (first) rank.

 

 

 

Te also have from (£1)
fungi: {577“‘P-

Thus the tangent to the curve v = const- on S! inte‘

""
(D
(J
C)
ct
U)
r F
’3
(D

line joining n t0{9 in he covar'ant pcin

fianF:(f, (3/01+Lg),

Therefore the image in S5 of this point is the line jtining
fix) pnl+ijo This latter point must be a covarient noint
also. We shall investigate the geometrical sigiificince of
this point.
We have from (18)
Yum—41¢"; Pt: flfivtl-‘J
= @(Ab-azfi)+”5fi'

Therefore
15w... “dlw’(l"(32;,)fi = (awe/+l- ‘3'

Thus the point 61°+L % is the intersection of the osculatins

*crove, Nets, p. 485.

' I

‘*Ibid, p. 4 o.

17.

Plane to the curve v = const. on S, with the line 14». ts may,

therefore, state the theorem;

The tangent line _t_g_ the gararnetric curve .‘I. = const. _0__ g; _a_t_ I
9;.

intersects the line he in g_ooin - The image 'n'g

 

point 's the line joining z,to the point of intersectior of

 

 

‘1‘ . ’n 4- 4-‘« 4‘ -* v‘ is . 1 _ {a A 4- fl
the osculatin;_plane .. t-e gar metric curve 1 — cons.. n 3,

 

 

 

at Z_Tith the line %fl3
Similar theorems may be stated for the tangent lines to the

Q
U

0)

parametric curves on Sh’ f, 5.
From (Bl) we have

fv—(m-;)f=(b+£)7fM/°~£J,
§~-4 ; = 6g 7 + Lfk

y‘

° 0A1 3!

‘ 4—
I.)

U)

If L # O, that is, if the parametric curve v = con
is not asvmptotic, we may eliminate/0 between the the equations

above. he resulting expression is

fv*”+t—“—(a.—3"i[&)r= (b+£‘%‘p)}(-L.

Thus the cevariant point
(’b4-fii"§?@)77— is
in 85 lies in the tangent plane to S; at {and on the line #4-
Its image in :3:5 is the line with coordinates
(bug-i, ‘ Q—a)(2’,¢,)-(/L)A,): [,t-(tn- i; “344)“!!ij
= [)2...‘<b~ “FHA/JIJ

The image is, therefore, the line joining 4.to the ray"pcint
on the line 3' of the point it trit‘n respect to the net 1:, . We

may state the theorem:

*Grove, Nets, p. 483.

18o

"‘d,to the r83 scint of

 

 

the Leint g_“1tn rec‘sct t th= net “I is the poi t of inter-

 

 

 

section fient plane t 5;— at E Wifl” +1“? e1 5M}! ___.h

0 l
H, H,
H

(D

p—h

(D

H

(I)

:3

0

CD

 

nyrewid
We may find similar results by using the tangent planes to

H

"a- ..
8)), D! , On.
We also have from (El)

hw-(b'£)?-Mf =(a-+2;)€ 1%,.

Thus the plane determined by the point

to
.8
F
,3’
.0
t“).
:3
O)
L .
P
:25
cf-
(D
H
l

sects the line $1,, in the point

(CL~+%§ f +—&b
The image of this point in 33 is the line joining.A M) flv—ti'.
The latter point is the intersection of the line in Green's
relation* R.to f withvj. Likewise we may find the po nt in 85
vLich is the inab e of the line in 33 joining s to the inter-
section of the line in Green's relation R to p with.f. Similar
results may be obtained using the expressions for )7“, and EV
We may state the theorem;

The plane in 8,- determined by 21...: 21., F inters ects the line {g
‘—'-b

 

in a point. The image 9: this point in §3 is the line joining

to the inter sec ction of tlm line in relation 3 EEHfi

p.

 

 

 

with n,
A, front (31‘ fife a:

or

Usin; the expressions for {w an
eliz; inate in t11ng’,;7 , {,fi provided II ¢ 0,1275 O, I? 7‘ C, I: 7! Co

A

We may thus obtain eight eXpressions or the form

*Grove, Nets, p. 431.

 

7; -ME LIV ‘
'jfilv 77’" M7, Nfo,

[junk/,1, r]

 

N
E—M“ ~M J
Thds in By the s3 determined by the tangeit plane to h and the

point t intersects the line joining77 and i’in the point
.~h N’ U +"”'D’_' h '
This point in 85 corresponds to the line iL 83 joining,¢ to the
point
1.. N—mm_%+ LN-M‘g
.nm N' A/
The latter is a covariant point on the linefao We shall leave

the geometrical interpretation of these points for some future

. H ': ~ . . , . ‘»
date. If m = a = O, that 1s, 1f tne given nets are F ransforms,

4

pression (22) reduces to

Lie) Efiw,£v,£,flzf tat—«Hag u

we may mate a tr 1;::or“°tion

Y

J

H A (D
‘\'1

H
.r

.01 11‘.

m

the given nets are F trans

PI)

0

f the form 14:.- Af such that it will make /=;:' F]:- 3 :.- 0’
. The first of eq deti one (18) be cones

Yaw—4 ww—fikfii— 14m 3.

_r
f":

hus if the given nets are F transforms the intersection of the
three space determined 137,4»:{wz’ 3' with the line 7727 is the
point of intersection of the ane determined by K... , z“ , 12‘,

with the line? .
2. Second Derivatives

* LoP. Eisenhart, Transformation ,f -n“faces, Princeton University

 

 

Press,(1923) p. 34.

A I

We may also 21nd second. derivatives Off ,1‘ , )7 etc. with

res>oct to u and v. In pa rt101lar \e fin

,0“;- (fl- 1:),olr [Hm‘rn—nL-fi fiz?+P- til/3+(m14mm);
+MF);-(,4+1&+A){—fi}7

f,v=(B-9c)p1+tzava— 13%” emu/111111));
+mrE—(B+2;+z)>7-Y{,

f“: at fw+(dw+fim+€-§;rmL)E+(51,—{3£ +{3111)77 *flL

(23) —LE1-(@M+4L«g;§+L1)f,

p1,: (B—é;)p..+£ 614+MM- %%-“4.15 +6 “139me
+(mbv‘fitmfl’7HquH‘;

few-4 AfV +(dv+(37)‘f +(G‘Tr/3V+L»w) 77

+(flm-f—L I3~—’:—,§ +Lv)f’ '"L’7o
Using the expressions defir in: f“ and /» fron (21) we nay

elin inate‘v and § £10111 he first two of equations (33) obtaining
the following eXpressions

,owu )Fw+()fv+()/0=(/mw~mfl’/)f+ (mums: )7,

Fvv+()(3w+( )[JVHHH (44.,- 4~8~g “7+ (m'w)>’ 5',
wherein the coefficients Of,%.7m_af are immaterial for our

:0

purposes. We may readily see that p .1. or ,0W will be a linear

raietric nets v= const.

:1:

combination of WW , f’v 'f that is the p

or u = const. on the surface SF at p will be asymptotic if

CaseI ma—mH—feo, A9“””="a
Case II ,mv-mB-a—eO, ”9-41.47
Case III mw~MH“/= 0, f3 :0:
Case IV 41, V _ ,y, .3 ‘3 =0, 7 = a.

If n-m = O the nets fly and 2-1 are radial transforms 1". We

fix .. (-1
* ul‘OVC, Lets, n. 41.6.

‘

3

 

- 1 5 j- ' r . - , ‘ ,-' - ». ' v. _, .0 J-L‘ . A“. 'Y? ‘1
eiclnoed this case at the DEfilJnlmb 01 the paoe1. as shall
consider cases III and IV. he asymptotic nets on 3r ani 31.,

the focal surfaces of the coniruence of lin

V

(D
U1
‘5
U
C),
H
(D
C‘-
(T)
"I;
{.J
8
Q:

by the expressions'
(g ‘40., +MB) ab»; ' (M'fl’w) fidu”:o

(f— ”aw/mg) cat’- — (Mar/w) 7-1411 2 o.
For case III these equations reence to

JA~1=0 4/0’7-0.

J

(0

g“

H

H)
'1)

O

(D

U.‘

(I)
SD

A)
U)
1

Thus the asymptotic curves on the focal
form only one one parameter fa ily, is. t
are devzlopable. Case IV gives the same result. We may, there-

fore, state the theorem;

 

 

 

 

 

 

The parametric nets n.§P will ‘3 agygptotic if anl only if he
focal surfaces s1 and s1“ are developable-
W nag eli 1nate,¢ fion the t;1“C of equations (23) by

using the value of {,from (El). Te find

:11, +< ) t1,” )5 +1); = (apt: +p1.))7+(1,e~1+17L—z1{ flap-LE.

u'

€1.11. will be a linear combination of ’6‘“, 5: f if l3:(_= 0° Pica-"—
ever, we must excluce ttis case since it reduces
equations (El) to f~=¢ig . This equation infioates tLat the

surface 3; is only a 3UTVG° ffifimnflfi also b: a linear con-
0

. , H N . 5L... .--,. “1 -' H-‘“1
'e surface a »1-leo e OJbCln s1m1lar *esalts

. _‘. M .... _ _ “.7 ., .
1n1n1 g,,. ,x31rrx, ther11013, strte

(or

 

 

{Jae
'L"3 “1,". “"‘
LLu thao on;
The r31_:et:ic nets on 3? 09? ot be PS“‘“tOfiC-
e obta1n t1e s~ve result 10: the su11>c~s on, Of, :5.
$7.: 4.“ ,q, .: - 1 .9: ..' .. “1 -. C \ .._. . -_. .—‘ :._..'
.s1nc one exo1ession ce1lnin5 g; Iiom (kl; we ”a, ell inate

from the third of eq Hat one (85) if L # 0- Te ootai: the

P

folloving result

€11.11.+()€u_+c)§+()fz= fl /L-L f.

C1

‘s
1
‘¢

T‘ s a covariant po i‘11t on the line joining,fi‘h)£’in.3~ lies in

ct

he 33 determined by the osculating plane to tne curve v = const.
and the pointy). To th s point in 35 corresponds the line in

S joining; the point ,4, to the point fl‘L'I'Lfl . The Leo ne trical
inter r.) retation of the latter point was given on page 13 of this
panor. Siniler points may be found by using eXpressions involvin;
y)“, ’{u ’ va .

The functions'fi and € may be eliminated from the fourth of

eouations 85). The resulting err ~3ssion for pk, is of the forn
Pam-k WW“ 1?,“ >p=<m—MH1—(11-m;<b+£, 177 HAM/w») (Mia) g.

It is readily segn that the paranetr‘c 1ets on Slacan be con-
jugate, that is a linear reletion can exis st be tueen fay: fl~’fy ,

(°if and only if n~m = Co This is the condition that U, and K

‘3

m

 

 

 

be radial transforms which was mentioned previously. he may,
t1 erefo e state the theorem;
The pew netric 11stgg.§p cannot b; conjugate.

Eliminat 3:); fr 011 the last of equations ("3 ( pro)

no
we 11nd

fwv+ufw+()€,+cm = [gm/+11% ‘7£+Lv'g—(fiJ+/3,+ML)]f-Li,

Thus a ineer relation exists betwe;n. f1“ f1, iv and if and

only if L = W = O, that is the nersmetric net on 3¢,is asynqtotic.

 

 

 

 

 

only 1 the par“ stric nets on st are eey;ytotic-

 

Similar theorems may be stated concerning the para etric r1ets on

and their Relation to th

he line jci ning the points f andwq in 3.. Any

(#9) '7' = f + A7]_
The coor‘inates of anv point on the tangent line to the curve

traced by th: poi nt defined bv the express;on (74) will *e of the

SF

11+;vfi11m1m,g +/(u,>7+ “Ht;

H

[4+(m§;)%1£+f@+xi+(e+fi+x)ggun
+[L +AM +(M1'AN)§£]F +(A-7g2j“

If the tangent line coincides with the line joininfi f to n , that
is if the surface generated by r is a develOWa ole, the coefficients
of,: and L,in (25) must be zero. In that case #3 find

L +x(ffl+-(hA+-K N’)%fi,= 0,

A—éfi‘ :0.

A
[\‘1
O)

V

Elinineting A from equations (28) we find
(27) LIL»; rammw+nue=a

Liketise elimi11ati Mgfi,fronv (??) we find

F J
“’5
O

(2.3) L.+ ZMA +N/l’20.

w

Thus tLe line joining ted points ? ana & in S, gene“ates a

C“

ears eter faeelies of develoge*lc surfaces. From (22) we see
that the develonables of this con ruence correspond to the

0 * ' 10v ‘ ..’.‘ -'—- ‘
as; ptotlcs on St 1n 8 . ae may, tnereJore, state tee tneorem:

The line joinigg»§_and A in‘g generates §_Conqruedceo Ibe

 

 

 

 

 

 

* — 5
1 €:e n.§5 f the agyrnto 108 3 §_ '; gs are the curves 0

 

"\ - r‘,* . ‘1‘ t .‘g/xyh‘ q ‘1‘ . q , I J? -T.‘ f
age 3 muleu celreeéeua t t.e Qevelqgaoles o; the cei,ruence

 

A SiILlar theorem may be stated for fZ'and S .

I h“ -f.
v d-

'

 

0

v0
va" "
I '.'..L

 

art, L'P’
Princeton

..
C“
5“;

f‘

.4
“duh-.1

_

ews
cel

 

 

 

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