'E‘RANSEORMATTONS OF CLIEV Es

AND

NETS DECEEVES :N THE P‘ ENE

P TM TN TEE EEEEEEEEE

$21637 Bagiey

1934

           

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ACKNOWLEDGEMENT

In appreciation of the suggestions
and inspiration received from Doctor

Vernon Guy Grove.

TRANSFORMATIONS OF CURVES AND NETS OF CURVES
IN THE PLANE
A Thesis
submitted to the Faculty
of
MICHIGAN STATE COLLEGE

of
AGRICULTURE AND APPLIED SCIENCE

In Partial FUIfillment of the
Requirements for the Degree
of
Master of Arts
by

Inez Bagley

1934

TRANSFORMATIONS OF CURVES AND NETS OF CURVES
IN THE PLANE

1. INTRODUCTION

It is the purpose of this thesis to study in some
detail certain real point transformations of curves
and nets of curves in the plane into curves and.nets
of curves*in the same plane. We first set up some use-
ful formulae in the case in which the transformation
is a general point transformation. We then specialize
the transformations in various ways. For example, we
consider certain transformations which we shall call
E transformations. If we further specialize the
transformations we find that the transformation is a
transformation by reciprocal radii.

Let the non-homogeneous coordinates ‘1’, , Llof a
point P; ‘be given as analytic functions of two
variables uuwr'o The locus of’ Px is a net of curves
in the plane.

Let the curves (4,: const. and v 2 const. be the
lines respectively perpendicular to the x;—- axis and
parallel to the 'xn-axis. The parametric equations of

x, any therefore be made to assume the simple form

‘*N- G- Grove, Contributions tg_the theory g£_trans-

formations 9£_nets in_a_space , Transactions of the

 

American Mathematical Society, Vol.35, No.3, pp.683-688
Hereafter referred to as Grove, Iheorngf transformations.

(1) )L,:u..,x,_=v',
It follows therefore, that

Kn”: I) L,V=o,
(2) wazo, xxyt-I,

wa20, XKV = o, L”:

Let Pry be a point in the given plane- It follows
that its coordinates A?” 1Lare defined by an
expression of the form
(3) A'2).'=.7‘-+'97Cu.,-I'¢I‘5,..
wherein e and ¢ are arbitrary functions of u, and \r'.

The purpose of this thesis is to discuss the
transformation of curves and nets of curves in the

plane by the transformation (3)-
8. TI? TRANSFORMATION E

A transformation will he said t_o_ he an E gagg-
formation* g a_n_d_ 9_n_l_y_ if the pp_i_r_1_i_: g intersection
Q; the tangent $1.153; 33 the corresponding curves i_s_
egually distant £19m the corresponding points PX gn__d_
PI? ,

Let the tangent to the curve \r:Ir(ugthrough the

point P2! pass through the point 7. , whose coordinates

 

*Vo G. Grove, The transformation E gusts,

Transactions of the American Mathematical Society, V0.33,

No. 1, pp. 147-152.

are defined by an expression of the form
(4) i“- x+/1~,(Xa,+/\x,)

wherein

1_W_=A,451L :F..+/\x,,
Ana. Ax~

 

 

Likewise let the tangent to the corresponding curve
through the point P.‘ pass through the point EL
whose coordinates are defined by an expression of the
form

(5) 1»=*3*M~(~x~+’\fv)-

Computing the derivatives of fit with respect to u,
and \r , reapectively, and making use of equations
(2) we find that
‘t*‘( I+ei)x..+¢., xy,

(6) Aav=<|+¢.)x,+eyx..,.

By substituting in (5) the values for 43‘” and .3

V

given in (6) we obtain

(7) 2‘: x +{9+/~i[(|+a..)r A94} X... + {¢+,w,_[¢..+/\(l+¢,)D 1...
Let P: , coincide with Piaf From (4) and (7) we

obtain the following equations
I“: ‘M"[(|+61~)+Aey} =9;
(8) AM! ’M»[¢~*/\(l+¢vu 3 ¢ ~

Solving (8) for ,u.‘ and #Lwe obtain

,u.. = ¢ECI+eul+A0J~9|Z¢a+A014.”
(9) (¢‘~ *Adi) -A(o~+/\e..) J

Mt: LL'¢
(¢..+A¢,)- Me...+/\e.)

(¢a+A¢y)-A(0wtdev)¢o.

Let the distance from 2, to 7‘ be 0L, and that from

Z

I.

to it be at» . It follows that

is =¢~."£z:x:+..>\u..x. + #2131.

1 s
(O) is. =P:[£;aw+t_}\21m1V+A‘Z‘ar‘]-

If we define E , F , G and '5’ , if, E respectively, by

the formulae
Eezxw‘,F:Zx-x,,G:zx,‘,

(11) - p ._ ..

E 12:13... , F =Zga1v,6=£‘a,‘,

it follows from (2) that

(12) EZGzl’F‘LO.’

and, likewise, from (6) it follows that
E:(I+e..)"+¢~.",

(13) E 9v('+6~)+¢-U+¢.).

a 3Ll+¢y)L+’ey‘.

If our transformation is an. E transformation it
is necessary that
(14) (A,:OL,_
After some computation involving the use of equations
(9) to (13), inclusive, this condition may be written

{[9 0"- "¢U*9-)] " AE9<I+ ¢..) -e,p]}"( H-A") a

(15 ‘ ‘_ ‘
) (As ~03 {U H e-)‘+¢.:]+ MEGAN-6...“. ¢~( )+¢,)+A‘[(H¢_) +6.3.

We may now consider (15) as an identity in .A and

equate coefficients of corresponding terms. Hence

[6¢...-¢( H- own": ¢‘[( ”0.)" + 0):] ,
[ec u+¢.)-ev¢J[e¢--¢u+e-n -.
“(MUI+a.)‘+¢..‘}+¢‘£¢-(I+¢.)+e,(:+a.;JJ

[eu+o.)-e,¢]‘+[e¢,-¢u+o..)]*=e‘[<wean as]
(15) ~Ve¢[¢i(:+¢.)+s.(|+e.,)]+¢1~[(I+¢,r+as],

[9(I+¢.)- e.n][e ¢iv¢(l+e..)]=

6‘[6.(1+0-)+¢.(I+¢.)]-e¢[(¢+o.)‘-+ 9:],
[60+ o.)~e.¢1"- 9‘E(l+p,J‘-+9J’].

The first and last equations in (16) may be written
in the following simpler form, respectively,
(eh W) (b... = z 9 ac ”6.),
(17)
(o‘-— eve. :15¢( |+0.).
By combining the two equations of (17) we obtain
(18) U+e..3e.,+(:+¢.)m-=a.
Likewise by using (18) and the second, third, and
fourth equations of (16) we obtain
(19) (1+ 0 .)"+¢.: ~. ( 1+ a.)"+ 6.."-
By combining (18) and (19) we obtain the equation
(6..h ¢..")[U+ ¢.J‘+e.‘l=o.
Since we are restricting ourselves to real trans-

formations we get from
(|+¢v)‘-+ art: 0

|+¢uzo)ev:0
Therefore

¢= -v+'U',(u-), 63- Vt.(k')o

It follows that
43.=*3.(-);*§» =1..(w).
Hence the locus of 9.? is not a net but a curve.
We will therefore restrict ourselves to the case
wherein
(30) 6.‘-— 46:.
From (18), the third equation of (16), and (19) we
obtain
(e‘-¢‘)(¢..‘—e,~ :0.
The second factor in this equation leads to the same
conclusion as (20). Furthermore
eho‘ia.
The existence of the above inequality may be demon-
strated by assuming a ‘- 0‘ = o and solving (1'?)
for e and ¢ . We obtain
9=~u~+¢,, d:-v+Co..-
Substituting these solutions in (4) we find that
18,-:CQJ A%,_: (LL.
Hencemouwm .memi‘mi 9, 1222391
22323; By using (17) and the fourth equation of (16)
we find that (20) may be written in the form
(21) e,-..¢..,
From (17) and (21) it follows that
(33) 9~+¢.+L=o.

It is possible to show after some computation that
the various equations in (16) to (22), inclusive, are
all satisfied if e and a satisfy the following
equations

a)... = 9.,
(23) 9w+¢v+1=0,
(e‘- ¢‘)6.. = 2. 9;“ 1+0...)

Thus we arrive at the theorem:

The transformation defined by equation (4) will b_e_ _a_

 

transformation E _i_f_ and only if the functions 9

 

 

and ¢ satisfy the system 3: differential equations

ma.

3. THE RADIAL TRANSFORMATION R

We shall say that g transformation gt; _a_ plane into

itself 13 g radial transformation if and only if. the
lines mining corresponding points 1.5; the plane pass

through 3 fixed point.

For example the coordinates of P, are (a, w)
and the coordinates of P? are (awe, v+ a!) . Hence

the equation of the line 9, 9.? is

 

x. A l
“a v I :0.
one v+¢ I)

 

This line passes through the origin if and only if

(24) W¢=V6.

Hence a_necessary and sufficient condition that our

 

transformation b§.a radial transformation ;§_that

UL¢=V9.

4. A TRANSFORMATION WHICH IS BOTH E AND R .

If we impose the condition that our transformation
be both an E and an IR transformation it is necessary
and sufficient that the following system of differential
equations be satisfied

u.¢~.ve,¢..:.6r,
(25) ew+ay+1=o,
(6‘-¢‘)0.:ia¢(l+e..),
However the last of these may be disregarded since by
computation we may show that it is satisfied if the
other three are satisfied.

We shall now proceed to solve for s and ¢ .

Differentiating the second and third equations in
(25) with respect to x» and tr , respectively, we

obtain

¢~w=e~VJ¢mv19vvj

(36)
euw+¢wvlol 6wv+¢,, :0.

combining the equations in (26) we obtain the Laplace

differential equations

evv+6w-=OJ

¢~.w+ ¢..= 0-,
and, hence, know that a solution does exist. From the
first and last equation of (25) we obtain

(27) (kt-v‘)6v=1u,\r(l+a..).

From the theory of differential equations we know that
the solution of (27) may be found from the solutions

of the following system
$4... -_ £3; 2.4.6

 

 

 

 

1W U.“U'L ~1w
If we solve the equation
a... 2.419..
1w ~1W
we obtain
(28) w—ffl 7. Q,
Also,
L :- (L‘U
V‘-W" 1w

This differential equation is homogeneous and may be

solved by the usual methods. We obtain as its solution

(29) W‘W'V‘ 1C;

W

Hence we know the general solution of (28) is of the

form
(80) e 1- u. -. FCw/ua‘)
wherein a. I

M, 3'

“’1.va

and F is an arbitrary function. Let 1 :. 9.709".
Differentiating the first equation of (25) with respect

to - and substituting the result into the second, we

10

obtain
YounweyzLa.
W

Likewise, differentiating (30) with respect to u. and

\r we obtain

(31) ewzF’Q—E-IJOVZF‘AE.
5%- by
We now differentiate 1- with respect to u. and u- and
obtain
3% -.-. 9. WM/‘hu. +,uu‘)
(32) a “’
bi: ”=1LIx./u./Awr.

av
By combining (31) and (32) we arrive at the conclusion

that
FzK‘ w/(Az‘

whereinl( is an arbitrary non-zero constant- Hence
6 =(K‘M‘- Uh.)

¢ 2. (K‘yy‘-I)V'-

Hence necessary and sufficient conditions that an

(33)

E transformation be a radial transformation is that

9 1 (KW‘ul) u.) ¢ ‘(K‘p‘—l)v.

5° THE TRANSFORMATION BY RECIPROCAL RADIIo

Let there be given a circle whose center is <3 and
radius Lia - Let P, and P,’ be two points collinear

With 0 811011 that
(3%) 3?. X 5—5 : Lt.

11

The relation existing between points P, P,_ satisfy-
ing (34) is called the transformation by reciprocal
radii.* The point 0 is called the center 9;;

fi-
inversion, the given circle the circle _o_f inversion,

 

and its radius the radius 9; inversion.

Let the center of inversion be the origin. We now
further consider the radial transformationHm E dis-
cussed in section 4.

If re, and A‘denote the distances from the points
P; and P1 , respectively, to the origin we find

readily that

__-

IL, 2 \(u‘rv‘ , ILL: Ww-fej‘flv‘lw)‘ .
Hence N1/DL:KL
(55)
Furthermore the coordinates of P1 are
(36) 43,2K‘wp",*).1.=K"V/u«".
They may be eXpressed in the form
~ K"x
A? - I. x I.
which is the formula of a transformation by reciprocal

 

radii. We note that the transformation (36) may be

written in the form I?) -. XfO 1., + ¢ 1 ,. wherein a

 

Eynder and Sisam, Analytic Geometry 93: space, New
York, Henry Holt and Company, 1914, pp. 201-3.
“hi—Io Lo Coolidge, A treatise on the circle and the

sphere, oxford, At the Claredon Press, 1916, pp.21-2.

 

*fihrove, Theory 91 transformations.

12

and ¢ are defined by (33).

Hence -i_f_ the points P, 93 the plane are trans-

 

 

formed by g transformation which i_s_ both E. and

 

 

radial into points P‘t 9_f_ the plane, the points P? are

the transformation _o_f the points P, by _a_t_ transformation

 

 

by reciprocal radii.

 

6. THE TRANSFORMATION OF NETS IN THE PLANE

In this section we shall set up the system of partial
differential equations by means of which a given net
in the plane may be transformed into an arbitrary net
in the plane.

We first make a transformation of coordinates from
the original coordinates

X5 = “e,'¥t e U'

introduced in section 1 to coordinates §,11 by
means of the transformation
(37) $=Y(w.v).\1'~h(‘~.~rl.

Consider the differential equation of a general net

(38) (Av-AL)(L~-BJ~):O
wherein
(39) A:A(\~,V),B:.B(te,u-)

are arbitrary functions of the indicated arguments.

Differentiation of (37) gives

13

(Ly :waLu-‘kgvd",
dun :.w1u.Aax.+-V(V¢L~r.

By a proper choice of ‘g and ‘W. we may make the new

(40)

parametric curves 3‘ =. const, v! - const~ be the
integral curves respectively of

AIL.:.B¢Lep, AA! =.A.L~..
It follows that

A: :f(A~..—B<L-)=OI

(41)
Anar(dflr-AM):0.

wherein /: and a— are factors of proportionality-
By comparison with (40) we find that (41) will be
satisfied by choosing
(a) fv=~8$»,we=-Anv-
We h8g3 £1129. made j:_l_l_e_ ggneral 2.213. parametric-

In addition, it is known that a parametric ggyngg
3g; m*;_s_ m conjugatg egg asymtotico Hence
the coordinates of I: satisfy the following system of

differential equations

wazfX'l'O‘Xua‘i-Ielr)

(43) XhV:CX-‘FQXW+J’¥V)

1yv=%X+qu*SXV-

 

*R- P- Lane, Projective differential geomgtry gflcurves
and surfaces, University of Chicago Press, Chicago, 1932,
p0 133-

14

The case wherein I~AB 3 a must be excluded, for

if this were true, the net would degenerate into a one
parameter family of curves
(M— BM)L:O

Combining (2) and (43) we find that

1a.: tag-.10

<44) 1‘ ’
flzfl : a:L:K:C:o,

If we define H and K by the formulae

H=C+a.lr— a...,

(45)
K2-CI+O-L"‘Tuv

it follows that

H=K

Hence the net Nx coggosed 9;: the lines (1.: const.,
V': const. has equal point invariants.

Likewise the coordinates of “P will satisfy a
corresponding set of differential equations, namely
4a,“,ziaafi +£1.A&.e-+j?1}.,
(46) aauveaek+iAkat 1:1,,

*VV $§1+Ffiw+§fiyo
Solving (6) for l... and X... we obtain

R1-=(|+¢u) u—‘¢u.. v
(47) w *2 ,

RXV '—(|+6w)“bv ~6vkaw.
wherein

R -_U+6.~\(i+¢,) — 9.,(0... it o].

15

because if this were true, At. would be a function of
one variable which is contrary to our original hypothesis.
Calculation of the three second derivatives of at,

with respect to u. and \r'and substitution of (47)

gives

Raquel] Howe-.. -6.¢....].~3..

+[(a1—s...)c.~..—¢e....] A}...

R1.~\r :[(_(+¢,)6...~6v¢u.u]*&u-

(48)
+[((te...)¢...e ~ ¢u9ue1~3 v,

R layy :[(I+¢yjeyy ‘6' $0911 “D

+[(’+9*)¢9v“¢u—6vv1‘35’ -
It follows that

7., :2: e I :0,
R Z = ( l+¢.36--~e.¢e-.
R; r. (l+9...)¢..,... -¢...s.......,
(49) «a
RL (lieu)¢wv'¢wGKV,
R37 —. (n+q,)e..-— 9. s»...

R‘s. : ( ‘1 a~)¢vv “ ¢w6.V.
Let us impose the condition that the transformation

( '1' ¢v)euv ”ev¢u.\f,

(I

be an E transformation. By combining (15), (23), and
(49) we find that

Izlrz~§=P

)

a:-~=—s—:.Q

16

wherein
: 1. E
.ai‘l
It follows that
Irv,- iwzo.’
and
F R“

Hence §_sufficient condition that the net Iva,

 

cogposed p§_lines respectively perpendicular and parallel
pp ppg_x-axis have equal point invariants i§_phap Egg
transformation p_e_ a}; E transformation.
Differentiating an arbitrary function I. by the
rules of the calculus we obtain
Zip :JC,'§..+dZL‘n.UI
JLV==1CY‘§,,+-1Cn‘1y.
Substituting (42) into (51) we obtain
1...:— ‘wa ~A ”VI
I a =— 8;..1, +H,x,,

(51)

(52)

Solving (52) for JCT and II" we obtain

 

Xy‘z-xa. +AI,
(53) ?w(|-AB) ’
2X.“ zly+Bx..
3"( A'A‘B)

1?

Calculation of the three second derivatives of]:

with respect to w and \r gives

pr‘rw‘xfl’ ‘ 1" YK‘I-rxf‘l ”‘1"an

+3....“I; ‘(AMWV+A~H V)X1‘ ,

(54)X.,- = eefgx” .c ”surfing” -A aux.”

~45»)... +Bfw~1Xy+11equ,

XVV :Bxywxxgy”Lafw‘IVz-‘yn'fny‘rt'fi

'BVIWFBvaII)’ Inky—rt)

Since 1“,, I“, , and I“, are linear combinations

of I“, and I, we may write the following equalities

(55)

Xa~=¢,()'.~I$-A‘1VX.7)
+fl.(’Bywx$-+MVX'1))

va:¢'($|~Xr ~‘A‘1Vxn)
+’L,('ch~xf+ylvxh),
Ivy: X.($~X5 ‘ANVXHJ

t S. (- 8)....1" fnyxn).

By combining (54) and (55) and solving for If? ,

I)“: , and I“ we obtain

VJU-ABVXfi =
new... '76. an: r--)+M(B.$.+Bfw-+af~"~”M

+A‘L( By‘w" BTKV +rl Two-S' 87%)]Iy

14th Men. ~o<.A n. +flm.)+m<4.n. «mam-nu)

+A"($."\y ‘rlAHV-‘V‘ VV‘]II1’

18

TwWVCI‘ABr-xyw:
faked... #9,”...-f.,.,)+(i+ne)(8.f.+ay..-+a.,f-~t,sgg)
+A(B..)’..,+ 3y“... ., rd}; -S,By.~)]xy
+[B(An..,v+A-14,~..,A11,+/9.n.)+(I+As)(L,h.~¢,/Wr’lum)
+A($n\, - r, A»), —M..)]I., ,
(56)
31,1-(I~A emu”:
E B.f..,+ airy-Pr, 7“ ~ 3,87g)+lBLBaf~+Bywh+¢fw~tleyd
+-B‘(.<,$u,-/3,syw~f....)] 1:,
+£(s....—-ir./+npmmlBHero-anhw)
+B"(A men—A01 p~$.A V). +AM.)]I,,.
It is evident that I.” , 1..., , and 1C.“7 are
linear combinations of If and I1, 0 We find the
coefficients of I; and X11 to be
“’42. 3(a..'3,3)+1&(a..~ (”8+ B...)+A"(B.+3’n‘$a3+33:1’ 1...... ,
in PM)" “5‘
#1- ._._ [(A u..+A"A ,~oc,A-+/8,)+2.A(1r,-a.,fl)+fi'k(si“a’n’qu“'i
Y...‘( \-A 81"

one 8(aL-rs,8)+(l+AB)(a..~Jr,B)+A(By+63—S,B)+B.~
(57) \A \-AB)"
11-": B(A..~_,gi,+fg,)+AB(.(,-,~CL,A)+fl($,~d’,fi)+(1i-¢lfl+4vl)
fen-eer-

(I ~. ELB.+BB...+ r.-$.B)+LB(a—n-1r13)+3:(“v"‘9'8)]7°~ .
1\Ir"(\"'A BP— \

 

8.1.1 (.8. ~ 3’, A) +1- 8(13.‘ an +AV)+BL(AL~~AAV-"Affi') .12;

‘1’”

 

19

By combining (44) and (57) we find that the original
net N; becomes, after the transformation (57), an

integral net yxv of the system

 

 

Xss=°¢'xg+fl'x.,,
(58) XIV:¢'XS*"'I¥‘7.
Xfifizy'X§¥-Sllh
wherein
ad: lAB...+A‘B,,-A"BB., .
3'...(I-AB)‘
fi’:(Aw+AA\/)hy ’
fa‘U-As)‘
0...:fl: +AB,’ b'aflwfiffl’ I
VvU-A-B)‘ w l—AB‘
(59) )‘c J

x.=(av+BBW)§w I
Mv"("AB)"

5' - 1A.B+e‘-A... ~B‘AA,
W1v (I‘l‘ 3)”

Likewise by combining (49) and (57) we find that
the net N41 becomes, after the transformation (37),

an integral net N7, of the system
‘35; “"19 ti'wm

(50) 1g“ .33.“..343.“
"t‘n = 7337 t 3",.“

wherein

1':(P+BQ)+LA-(Q-BP+B~)+-fi"(Bva-BQ—BBu) — 1....
§m(n~asit 5*“

I

E':flhw-tAAv-AP-Q)+1A(P-AQ)+A‘CQ+A13)) y“,

few-daw- ’

51—. (l+B‘)(Q-AP) +a.’,
“VLA-A'B)‘.

 

<61)... ,
Jo :M+A‘)(P~BQ) +1r ,

In—(l~fl'51"

t‘ =_£BK+BBv-P-BQ)+‘LB(Q~BP)+B"(P+BQ)] g... ,
\y‘h-ABJ‘

§'-.(Q +AP)+LB(P-AQ+&,)+B¥(A..~fifiv'AP—QL-3L,.
Mv(\~AB)" “Vt

 

wherein P and q are defined by the formulas (50).
Suppose now that the net of curves § = const.,

~\: const, is an orthogonal net. It follows that

A+B=o . Under this condition the coefficients E.‘ ,

1' in (60) may be written in the form

R’=Q+BP+a—':J;_«§fiJ-fl63+m’,

 

(62) ”Mitt/P)
L, = P+AQ «PW: Ji—L 141: + 4r].
how) 3‘
We find readily that
a'-3L';aa'-a '
2H; ‘1 a 611

We may state our result in the theorem:

21

Any E transformation pf _a_n_ orthogonal net with equal

point invariants $3 2;; orthogonal net with gqual point

invariants.

 

Impose the condition that the nets IV, and N.)

be conformal, namely

.E..-_E..=.G_.
'5- 7-" C’

From (12) and (13) it follows that
0+6...) “+ o.)- .. c u +e.i‘+ 6.),
6.L\+6.)+¢.. (Ii-<6.) :0-
Hence _i_f_ 3h; transformation _i_g 92 E transformation

it is, also, a_ conformal transformation.

 

I

Computing 6' , F , and s’ for the transformed net

"in. we obtain

 

 

 

El:— I‘Ffl‘. I
f.."(I-AB“
F’z A'i'B ’
(63) fava—ws)
G' a 1+ 8" .
‘\."(|-AB)‘

Likewise if we consider the net M3. , we obtain

E'Ilefi F+AI§__,
{.34 |~Aa)‘-

F'=BE +(I+AB)E+AC ,
(64) YaMVCI-AB)‘-

C" :- B‘Ei-lBEi-E
"\.‘(| -AB)"‘

22

Suppose now that the net A/x' is an orthogonal net.
It follows that I'd-6 z o . Let ”A3! be an E trans-

form of N1: . Under these conditions we may write

(63) and (64) in the form

 

 

 

 

E ’—_ I .
§’a.‘-(I+A1)
(65) F' t. 0’
G! 2 l
V1y"(|+’4")
E' -. E
th(I1—A~)
#3, ~ 0
(55) I
5': L .
\VL( 1*At)

We may therefore state the theorem:
Any E transform _o_f a3 isothermally orthogonal net

_i_z_l_ the plane 313 gig; isothermally orthogonal net.

BIBLIOGRAPHY

Coolidge, J. L. A treatise on the circle and the

-*————-_

sphere. Oxford, At the Claredon Press, 1916.

Grove, v. G. §_general theory p£_nets pp g_surface.

Transactions of the American mathematical

Society, Vol. 29, No. 4, pp. 801-814, October,1927.

Grove, v. G. Contributions pp_ppp_theory p£_p£§p§-
formations pf; WEE. _a_. m S... . Transactions
of the American Mathematical Society, Vol. 35,
No. 3, pp. 683-688, (1933).

Grove, v. G. The transformation E pi nets. Trans-

 

action of the American Mathematical Society,

Vol. 33, No. 1, pp. 147-152, (1930).

Synder and Sisam. Analytic Geometry pf_space. New
York, Henry Holt and Company, 1914.

   

 

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