RECTILINEAR CONGRUENCES

By
CHUAN-CHIH HSIUNG

A THESIS
Submitted to the School of Graduate Studies of Michigan
State College of Agriculture and Applied Science
in partial fulfillment of the requirements
for the degree of
DOCTOR 03? PHILOSOPHY

Department of Mathematics
194$

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ACKNOWLEDGMENT
The author wishes to thank Professor
Vernon Guy Grove for his kind suggestions
during the preparation of this thesis.

CONTENTS

Page
Introduction

.

1

1* Differential Equations and Integrability
C o n d i t i o n s .............

.................

2. Transformations, Invariants and Covariants

•

3* Power Series Expansions for the Surfaces Sy,

4
.

7

Sg.

12

4* Canonical Form of the Differential Equations and
Loci of Some O s c u l a n t s ....................... 18
5* Moutard Quadrics ofthe Surfaces Sy , Sz

.

6 * Segre-Darboux Nets on the Surfaces Sy, Sz .

23
*

.

29

7. W C o n g r u e n c e s ...................................33
8 . Curves of the Focal Nets Ny,N z

................... 37

9. Correspondences Associated with the Focal Nets
Ny, Nz

...................................... 40

10. Axis Congruences and Ray Congruences
11. The Congruences z 7 , y$

. . . .

44

and the Principal

Congruence y 5 .......... .... ................. 48
12. Osculating Linear Complexes and Associated
Linear C o m p l e x e s ............................. 52
Bibliography......................................... 5#

RECTILINEAR CONGRUENCES

Introduction

In his prize memoir [11] (**■), Wilczynski has established
the theory of a rectilinear congruence in ordinary threedimensional projective space by using a system of linear
partial differential equations.

However, his method of

deriving the system of the differential equations is not
completely geometric.

The author proposes to remedy this

lack of geometric content in the present paper*
In §1 we introduce, by a purely geometric method, a
completely integrable system of linear homogeneous partial
differential equations which defines a rectilinear congruence
in ordinary space except for a projective transformation*
The integrability conditions of the system of the differen­
tial equations are also calculated*
In §2 we study the effect, on the differential equations
of § 1 , of a group of transformations which leave invariant
the focal nets Ny , N z on the focal surfaces Sy , S 2 of an
integral rectilinear congruence X

of these equations*

Some invariants and covariants of these equations under
( ) Numbers in brackets refer to the bibliography at
the end of the paper*

- 2 -

this group of transformations are also obtained and listed.
In §3 we calculate for the focal surfaces Sy, Sz local
power series expansions, each set of which expresses a local
nonhomogeneous projective coordinates of a point on one
surface as a power series in the other two coordinates and
represents the surface in the neighborhood of an ordinary
point on it#
In §4 a canonical form of the system of the differen­
tial equations of §1 is obtained by a geometric determina­
tion#

We then reduce the power series expansions of the

focal surfaces Sy, Sz in §3 to canonical forms, and find
the loci of some osculants associated with the plane
sections of the focal surfaces Sy , Sz made by a variable
plane through a generator yz of the congruence X •
In §5, by means of the quadrics of Moutard for the
tangents to the curves of the focal nets Ny, N z of the
focal surfaces Sy, S2 we study some special kinds of the
congruence X and determine geometrically the unit point
of the coordinate system for a general congruence X •
In §6 , we find the equations of the curves of Darboux
and the pencils of quadrics of Darboux at the focal points
y, z of the focal surfaces Sy, Sz and geometrically
characterize the congruence X when one or both of its
focal nets are Segre-Darboux nets.

The conditions for the

both surfaces Sy, S 2 to be isothermally asymptotic at the
same time are also deduced#

- 3 -

In §7, the Weingarten invariants and the tangential
invariants of the focal nets Ny, Nz are derived and a
simple geometrical characterization of a W congruence is
given.
§8 contains the local power series expansions for the

u-, v-curves of the focal net Ny on the focal surface Sy.
These expansions express two local nonhomogeneous projec­
tive coordinates of a point on each curve as power series
in the other coordinate, and represent the curve in the
neighborhood of an ordinary point on it.

Quadrics having

contact of different orders with the u-, v-curves and the
surface Sy at the point y are considered.
By a line lt[ 2[) we mean, as usual, any line through
the point y(z) of the surface Sy(Sz) and not lying in the
tangent plane of the surface Sy(Sz) at y(z); by a line 12
[ 1() we mean, dually, any line in the tangent plane of
the surface Sy(Sz ) at y(z) but not passing through the
point y(z).

In §9, we derive two correspondences between

and lz and between

and

1 *, and present another new

geometrical characterization of a W congruence*
§ 10 is concerned with the determination of the developables and focal surfaces of the axis congruences, and also
of the ray congruences, of the focal nets Ny, N z of the
congruence SC • The condition for the focal net Ny or Nz
to be harmonic is also obtained.
In §11, we study some covariant congruences associated

- 4 -

with the congruence X

by methods similar to those used in

§ 10 *

The last section is devoted to the derivation of the
equations of the osculating linear complex along a genera­
tor of a W congruence and the associated linear complexes
of the focal points y, z of the generator yz of the con­
gruence X *

1* Differential Equations and Integrability Conditions

Eirst of all, we consider in ordinary projective
space a congruence with two distinct proper focal surfaces
S , Sz generated respectively by the two focal points y,
y
z of a generator yz of the congruence. Let the parametric
curves u, v on the surfaces Sy, Sz be taken as the curves
of the conjugate nets N , N

in which the developables of

the congruence touch the surfaces Sy, Sz ; and let the utangent at the point y of the surface Sy and the v-tangent
at the point z of the surface Sz coincide in the generator
yz.

If we select two points 7 , 5 respectively on the v-

tangent at the point y of the surface Sy and the u-tangent
at the point z of the surface Sz, and if we suppose that
the coordinates

, 5 of the points 77 , 5

are functions

of u, v; then it can be shown that the coordinates y, z,
07 # ^

of corresponding points y, z, >7 , 5

satisfy a

system of linear homogeneous partial differential equations

- 5

of the form
yu = cky -t(3 z,
Zy^ryt/z,
yv = ay + c 7 ,

(1 .1 )
zu - b #z + e* 5 »
7^-my +nz+P7 + q
zuu = m ty + n ,z + p f7 + q f5

(cc,p tq/Sr^O)

in which subscripts indicate partial differentiation and
the coefficients are scalar functions of u, v.
The derivatives

, 7 ^ and ^

as linear combinations of y, z, 7 , ^

^

may be written

by using equations

(1 .1 ) and
(yu )v

^^v^u*

^^u^v— (zv^u*

the result is

I 7 ^ - ry + nz/c + s 7 tq ^ / c ,

(1 .2 )

- m fy/c’ + sfz + p* 7 /cf + r ’ £
7 ^ = e'yff'z + g*^ ,

in which the coefficients are defined by the following
equations:

- 6 -

C6

cf = /3v + fiS

cr = m -

- a^,

- a (3 ,

cs =p - Cv - ac,

g = cL - (log c)u ;
(1.3)
o ,et= r u + o ( r - b ,r ,

c*f*= <^U +/<3K -

c #r ,= qt -

c ’s ’^n* - b^ - b t2,

- b ,o f,

g* = <r - (log c*)v .
The integrability conditions of equations (1.1) are
found by the usual method from
^ ^u^v ~ ^
and the fact that the points y,
independent.

the equations
^^u*v ~ ^ ^v^u
z, 7 , 5 are linearly

These conditions are

ev+ae + f Y + gr - ru + roU-es + m fq/cc*,
fv + f<f+ gn/c = r /9 + (n/c)u +b*n/c -ffs t qst/c,
ce + gv = su + p fq/cc1,
gq = cfn -f-c(q/c)u + qr’;
(1.4) <

f^ -f b'f 1 f e ^ -f- g *s 1 = Sy + fa* -t t'r1+ p ’n/cc1,
e^+e'c^ -f g'ffi*/©1 = s ’y + (m’/c*)vtamf/ c ’+ e^Hj-p'r/c1,

c *f *+ g^ — r^. + p tq/cct,
g’p f= cm* + c f(p,/©,)v + p ,s*
Making use of equations (1.3), the third and the
seventh of equations (1 .4 ) we obtain the equation
(1 .5 )

(a + 6 * + <P + s ) u = ( g + b , + o( + r t)T .

- 7 -

It follows that there exists a function G of u, v which
is defined, except for an arbitrary additive constant, as
a solution of the differential equations
(1 .6 )

©u =g-f-b*+ c^-f-r*,

0Y = a + gtt / +

Accordingly, the following formula is valid:
(1.7)

(y, a, 7, 5) - e9 ,

where a determinant is indicated by writing only a typical
row within parentheses.

2. Transformations. Invariants and Covariants

Let us consider the group of transformations on the
the coordinates y, z, 7 , 5 and the parameters u, v:
(2 .1 )
(2.2)

y= Ay,

z=/iz, 77 = ^ 7 , Jr= rJ

u =U(u),

where A , ^

f

T =V(v)

(>yU.PT=£0 ),
(U*V**0),

are scalar functions of u, v and the

accent denotes differentiation with respect to the appro­
priate variable.
The effect of the transformation (2.1) on the system
of equations (1 .1 ) is to produce another system of equations
of the same form whose coefficients, indicated by dashes,
are given by the following formulas:

- 8 -

ot = o<. -

/a , ^ = ,6 /i/A,

y = y^/M,

a = a - A v/ x ,

c=ciVa,

b* =b* -m u/m , 5» = c'r^,

I =i;(iA-

. 2aAT +2>2/A),

n=n>^/^,

(2.3) <
P =-^(p - 2c A y/ A ),

q=qr/A,

m^m'A/^,

n ' = ^ ( n V ^ - ^ uu - 2b*/^u + 2 ^ 2/ ^ )>

V'-V' U /m >

5 , = J-{l’ -Zo'My/su).

The effect of the transformation (2.2) on the system
of equations (1 .1 ) is to produce another system of equations
of the same form whose coefficients, indicated by stars,
are given by the following formulas:
^*=<JL/XJ',
a* = a/T’,

(3 * = /6 /U»,

r* = x/V',

o* = c/V',

b ’*=b'/U»,

m* = (m - a V A ' l / V ' 2 ,

n * = n / V ’2,

p* = (p - c V " A ’)/V»2 ,

q* = q/V’2 ,

<f*=<T/V',
»*= c’/U',

(2.4) <
n’* = (n' - b'TJ"/U’)/XT*2 ,
p»*=p*/U’2,

q»*=:(q»

-

c ' U V U M / U ’2 .

From equations (1.3)* (2.3), (2.4) we may obtain,
after some calculation, the following functions which are
absolute invariants under the transformation (2 .1 ), and
relative invariants under the transformation (2 .2 ), of
the system of equations (1 .1 ):

- 9 -

A = cm'/p’i

A* = c’n/q,

B = m ‘is ,

B r= n r,

D = c*(6/q,

D' = c Y /p',

G = m ’n,
H —

(2.5)

—

K ’^ / j y ,

—

K = /3X ,

H*=

$ = p ’q/ec’,

ft *= p'q/ec’»

ft = c’f' + (c'n/q)v ,

§ ’= ee + ( cm’/p* )u >

^ W (u) = §

- K,

W (v )= ft‘ - H ’.

W (v) = ft - H,
I= C f//3

wtu)-*f
I ’= c’e’/ r y

,

J ’= o’f ’,

J = ce,
L = c’s’

^ v-(logK)UT,

L ’= cr,

y

M = 3a + g ’-2 p/c -hq^/q,

M ’= 3 b ’-f-g - 2q,/o,+ P ,u/pf9

K = n (2b ’ - q ’/e’+y'
P
- c ’n/q + qs’/n),

N *- m ’(2a - p / c t ^

_ P = 2a - p/c+^>v ,

-cm’/p* t- p ’r/m’),
P f= 2b’ - q’/c' -+■ y^u .

where
^ = log A,

~f = log A ’

The effect of the transformation (2.2) on each of
these invariants is given, with self-explanatory notation,
by the following formulas:

- 10 -

A* = (l/V* )A,

A* = {l/D')A»,

B *= (1/U'3)b ,

B .* = (i /Y*3)b »,

D* = (V,2 /U'2 )D,

D'*= (U*2/7'2 )D',

G* = (l/U,2V ,2 )G,
H* = (l/G’V )H,

k

**= (i /u »v *)k :»,

K* = (l/U'V’)K,

(2 .6 ) <

§*=(l/tJ'V») f>,

ft'*= (l/G'V) ft',

® * = (l/U'V') ft,

§'* = (l/G'T*)

W* . = (l/U'V')W,
(u)
(u)’
WfT) = d / u ' v )W(T),
i* = (i/v*)i,

w»* =(i/u*v*)w; .,
(u)
(u)
I'* = (l/U*)l',
j»* = (l/a’v )j *,

L* = (l/U,2)Lj

L'*= (l/Y'2 )L',

M* = (1/V* )M,

M ' * = (1 /U* )M’,

N* = (l/V,2 )N,

N** = (i/tr,2 )N*,

P* = (1/Y')P,

P»*= (I/O' )P» .

These invariants are obviously not all independent.

Among

them it is easy to obtain the following relations:
AA' = G/4>,
(2.7)

BB*=GK,

<j3- = H + I u = §• - Ay,
N = [L+A'(P* - A')J/D,

d d *= k /

^

,

J' = r + i ; = f t - A ; ,
N'=[L* + A(P - A)]/D».

In terms of the invariants, the integrability condi­
tions (1 .4 ) then can be written in the form

11 -

JV +J(P - <f>v ) + I K = L ^ + A ft',
IV + I(I + P - ^t )= L' +N,
K» -ft* =3(H* - J),

(2 .8 ) <

^ - r ’ - qu/ q = A * ;
J^ + J'(P' - t u ) + I'H’= L t + A ' § ,
1^ + X*(1»+P* - f u ) = L + N ' ,
H - § = 3(K - J'),
S

- s - Py P '=A.

Prom equations (1 .1 ), by differentiation and substitution we may obtain the Laplace equation of the focal net

V
(2 .9 )

yuv = ( c<v ^/3 y - oJ./-D()0v/(s)y + [fiy./(3 +<f )yu + a yy .

It is easy to show that the point invariants of LaplaceDarboux of the net Ny are H, K defined by equations (2.5).
Moreover, the point z is the ray-point y_^ of the v-curve
corresponding to the point y or the minus-first Laplace
transformed point of y with respect to the net Ny, and
the ray-point y^ of the u-curve corresponding to the point
y or the first Laplace transformed point of y with respect
to the net N
(2 .1 0 )

Jf

is defined by the equation
y1 = - iy + 0 7 .

Similarly, the point invariants of Laplace-Darboux
of the net N„z are H*, K f defined by equations (2.5), the
ray-point z^ of the u-curve corresponding to the point z

- 12 -

is the point y and the ray-point

the v-curve

corresponding to the point z is

(2 .11 )

z^=

- I ’z + c* $ .

3 . Power Series Expansions for the Surfaces Sy, S,

From system (1.1) and equations (1.2), (1.3) by
differentiation and substitution, any derivative of y can
be expressed as a linear combination of y, z,7 , ^ . In
particular, one obtains

uu

= ( <Xu + ^ 2)y +( (Su +^/S +b* /3 )z +o»/3 5 ,

uv = ( J.r +B.U +(3Y )y + (/3 y +{3<P)z t e ^ 7 ,
yuuu

°''uu+ 3 o U u + ^ 3 + m ’/3
+</3 u u + 2 c/u /8
^ / S u + 2 V /3U + ^2/3 + b* o^/3 + n* /3 )z -hV'fi7?
+ ( 2 c ’ / 3 u + c ^ /3 + q * / 3 ) 5 ,

yu u v :=(oi'uv'f'aoiu “f 2 ^ o(v + /^ur + /3
+ ac^2 )y + { /3UV t V v /3 +/9U
+ fizY)% +c(

+ 2 c<y6 X

+ o ( /6v ^ b ,/6 v
+cC2) 7

- f - c * ( /9v + / 3 <f ) 5 ,

Tuvv — (m^-Mnot + ep-hm’q/c* )y + (n^ + m/3 + b ’n -f-fp
(3.1)

<

-+qsf)z f(pu + gp -f^q/c* )7 -t(qu + c’n tqr f)5 ,
y

= (n^ -f-am+n rf-e*q fpr)y + (nv -f-n/ -t f *q+np/c)z
-f-(pv + c m + p s )7 -f- (qv -hg’q-f-pq/c) 5 ,

yuuuu = (*)y + (* )z +(3p,/3 u+Pi/« +2p'°t/9 - OuP’/9/®
+ P ’q' /3/o')7 -M3c'/3UU +3c* ^ u/3+2oV/3u

- 13 -

+ 3q'/3u +q^l/3 + c ,o(2/6 + c'n'/3 +
^

=(*)y+(*)y + rHvv + 21^ /

+

+n

)5 »

£rT + 2 f'qv + f'q

+ n p ^ o + (np/c)v.+ nin + ii cT2 + np //c + f 'q <f
+ f 'g*q + f ’pq/c + nps/ejz + ( q w +2g*qv + gj.q
+ PqT/ c + 2pvq/o - ovpq/o2 + g’pq/e + pqs/c
V.

+ mq + g ,2q) 5 »

where (*) denotes terms immaterial for our purpose.
The coordinates Y, where
Y = Y(ut^u, v + a v ),
of any point Y near the point y on the focal surface S
are given by the Taylor’s series
(3.2)

Y = y + yu nu +yT av + 4(yuUAU2 + 2yirrauAT +yTTAv2 ) + ...,

in which the increments au and a v correspond to displace­
ment on the surface Sy from the point y to the point Y.
If the points y, z, 7 , $

are used as the vertices of

the tetrahedron of reference, with unit point suitably
chosen, then any point given by an expression of the form
(3*3)

3qy + X2 Z

V + *4 ^

has local coordinates proportional to

..., x^.

Substi­

tution of the expressions (3 -1 ) leads to the following
power series expansions of the local coordinates of the
point Y:

- 14 -

7 2 = l + c ^ / i u t a A V + J ( «^u +cA2 ) ^ u 2 + ( o(y + a o C -f-^ rjA U A v
-f-JmAV2

-h • . • ,

7 2 = /3A U -/"i( /3u ^ 0(/6 + b ,yS)Au2 + ( /8 v -f-/3 S ) A X l A V f JllAV2

ff ( /^uu + 2

+ 2b V3u ^ 2 /3+b’0^ + I1V3 ^ u3

+ 4 < Z3 u v +'*r/3 +0

^v + b W

+ b ,/d<^)AU2A v

+ z6

+^ ^ Z 3 2 r

f ^ ( n ^ -mh^ t b ' n - /- f p -f-qs f ) a u a v 2

+ - | ( n v - M i £r + f ’ q - f - n p / e ) A v 3 t . . . ■ h ^ [ n w

+ ( n p / o ) v + mn + n<f2 + n p <P/ c

+ 2 f *q v - f f ^ q f - n p v / c
+ f ' q / + f ’ g’ q

-f-2nv ^n

+ f *p q /c + n p s /c jA v ^ + - . . . ,

(3.4) < y ^ = o a v t c c ^ A U A V + i p A v 2 + ^ ( /3 uU + 2 ^ u/6to^/s u + 2 b ’/6u
-h^2/3 t b U / 3 + n ’ /3 ) a u 3 - f . . .

~t~.

t ^ - ( p y +cm t p s ) A v 3

. . ,

y^ = ic */8 au2 + JqAV2 +

(2cf/Su + cf«t/3 + qf/3) a

+ i c * ( , 5 -/-/3 /)a u 2 a v + £ ( q u + c ’ n + q r * Jauav2

+ i(q^ ^ g ’q +pq/c) av3 + ^ ( 3 e ffl uu +3c’o(u/3
t 2 c U / 3 u + 3 q ’^ u + q ^ / 3 -^c’oC2^

+e*n9fi t q ’ oZ^

tq ’r ’^ A U ^ + ... f ^ l q ^ + 2g’qy +g^q+pqv/c
+ 2py q /c

-

c v p q / c 2 + g f p q / c - e p q s / c +-mq + g , 2 q )A V ^

Division of the last three of the series (3.4) by the
first gives the following expansions:
y 2 ^yl ~ P AU + ^

+ 1)1/6

-h(p v +/Sjx-a/3)AUAV

+nAT2/2 + f ( /3UU - <^U /S - 2<^S|1+ 2b,(3u + =<2 /S
- 2b'o<./3 + b .'/3)a u 3 + ..,
- 3an)Av3 + .

-jriiiy t n<f+f *q +np/c
+ 2n v ^-f n£fv + 2 f*qv

- 15 -

+ f,J.q - 4 anv + npv/c +(np/c)y - 5mn-f-n<T2
- 4 ancT + n p //c - 4anp/c + ffq <F + f’g'q
- 4 a f *q + 12a2n + f *pq/c + n ps/c ]

a

+ ...,

y ^ / j l = C A V t £ ( p - 2ac)AV2 + | p ’ p u ^ + . . . + T 'fP y

+ ps - 3ap - 2cm+6a 2 c)AV5 ^ ^ ( 3 p f/3U + p ^ ^
(3.5) ^

-SpU^ - cup y c t p fqtj8 /c,) ^

+ ...,

7 ^ / y ^ £cf/3AU2 tj^AV2 -f--^-Ue^ + q ^ - 2cV/S)au3
^i(of/3v -f-c9/3S' - acf^6 )AU2Av + i(qu t c ’n -qo*.
tqrf)AUAV2 + -g-fq^-t-g*q - 3aq-f-pq/c)Av3
"^2^-(^^f/^UU " 3ofc^y_|S — 6 c

-f3q,y3^1 +

-tc’n'/S+3cfoC2/3 - 3qV.^9+q,r ,/a)AU^'+(*)a u 3av

+(*)a u 2av 2 -+-(*)auav 3 + ^ :(qV v -hg^q+2 g,qv
- 4 aqy + pqv/c-f-2pvq/c - cypq/o2 - 5mq-f-12 a2q
- 4ag*q -hgt2q - 4apq/c + gfpq/c -f-pqs/cjAV*'
+

• • •

From the expansion (3*5) it is possible to calculate
to as many terms as desired an expansion for Y^/y± as a
power series in 72^ 1 * Y^/y^*

This calculation can be

performed by setting Yj^/Yi equal to a power series in y 2 /7 i*
Yj/Yi with undetermined coefficients and then demanding
that the expansions (3.5) for y 2/yi> Y^/yi* Y^/Y\ shall
satisfy this equation identically in a u and av as far as
the terms of a sufficiently high degree.

The result to

teims of the fourth degree is found, by the use of equa­
tions (2 .5 )9 to be

- 16 -

(3.6)

y^/yl =

'^y2/yl ^2/^ +

1^ 3/yi J2/2o2 + Bx (y2/yi)3

-o'I(y2/y1 )2 (y3/y1 )/2cp - qA» (yg/yj) (yg/y^
2C*/3
+ qM(y3/y1 )3/6c3+C 1 (y2/y1 )^+ ...
+ q [M t + M 2 + (P - ^ V )M +3L'J (y3/y.) 4/24c 4
~t *••,
where
B1 = (q,/3 +CV./3 - Sb'c'/S - o'fin )/6/33,
(3 .7 )

X

24/3^

(-c’/3 u u + c ^u/3+2c,M i - 5Vo*/8

+ q^/S- Je'n'p- c U 2^ t5b'cU/3 - 3b‘2c ,/9
-fq^*^ )•
The local coordinates z^, .

z^ of any point Z near

the point z on the focal surface Sz and an analogous power
series expansion for the surface S 2 in the neighborhood of
the point z can be obtained in a way similar to the forego­
ing, or else can be written immediately by making the
substitutions
/y>}> u l 3 c < / 3 a

(3.8 )

e

[
\ z ^ v 2 4 £ Y b* c* f*
A B D G H K

(3.9)

c

$

A* B» D' G K» H« ft'

The result is

ft

f

g m

n

p

q

r

s\

)»
e’ g* n ’ m* q* p ’ s* r’ /

W(u) W, > I
WJv) Wj

J

L

M

N

P

I* J« L* M ’ N* P*

- 17 -

-fJm’Au2 -f-( r u +oCrjauzw f£( Y^-trS + ax)av2

Z1 ™

+

^ e*P*+ m *Q.*/0*

+ i(my- -f-nfx

-t-am* -P©#q f -f-p*r)AU2AV + i( ruv +<*v x-f- aru + x u J"
*f*<^rv +rj'u +a<^r+otrJNY3 r2 )4uAv2 + -£( xw
-t2axy t yv <rt2 X<rv + ar<?+r<T2 + mrjAv^-y . ♦.,
(3 « 1 0 )

^ z2 = 1 + 1 ) , a u + J ' a v + i n f Au2 +( cPu + /3 X y V c T ) ^ u ^ v
+ i( <fv + S z )av2 + .•#,
Z^ - ip *A U 2 + 4 c X A V 2
+ i(p ^

-h

~ £ " (P ^ - X g p f

-f-

-^cm , + p f s ) A u 2 A v + £ © (

P f <3.f / c f ) a \ P

r u + o (r)A U A v 2

+ ~ -(2 c ry te X J ' + p /)A v 3 y ♦ . * ,

Z. = C 'AU -h^q’AU2 + cMaUAV -f ♦.«;
y

(3.11)

z3/z2

1*1/ z2 )2 y p f(z^/z2)2/2c 12 + B£ (zx/z2 )5
- cl*(z1 /z2 )2 (z^/z2 )/2otr« p»A (z1 /z2 )(z^/z2 )2
y p fM* (z^/z2 )^/6c f^ + C£(z1/z2 )** -h ♦..
-f-p’[M’2+M^ +(Pf - ^ u )M» + 31] (z4/z2 )V24c,lt"I" •••y

where we have placed
B£ = ( p r + c n T - 3ac y - orY )/6

(3.12)

C,»= —
-1

(- c r

,

+2cy / + cr<f - 5acr tp r

2 4 r^

- 3omy - orif^ + 5acy<f - 3a2 cy + psy ).

- IS -

k* Canonical Form of the Differential Equations
and Loci of Some Osculants

For the purpose of choosing for the points y , S

two

particular covariant points respectively on the v-tangent
at the point y of the surface Sy and the u-tangent at the
point z of the surface S2 , we recall the definitions T5J
of two osculants associated with two ordinary points of
the second kind of two plane curves.

Suppose that 0^, 0 2

ere two ordinary points of the second kind of two plane
curves C-^, C2 respectively, so that 0^02 is the common
tangent.

Let K^,

Cl* C2

Po5-nts

be any four-point conics of the curves
°2* and

P0^ar H nes

of the points 0 2 , 0 ^ with respect to the conics K^, K2 ,
respectively; then the intersection of the polar lines X it
lz is called the “principal point associated with the points
®1* ^2

curves

®2# Moreover, among the pencils

of four-point conics of the curves C^, C2 at the points 0^,
0 2 , there are principal conics which pass through the prin­

cipal point.
Let us consider a general plane n through the genera­
tor yz:
(4 #1 )

x^-Ax^-O

(A. 7^ 0 )

.

It is obvious that the plane yr intersects the surfaces S ,
V

S

z

in two curves C , Cz having y, z as two ordinary points

- 19 -

of the second kind.

By virtue of equations (3*6), (4.1)

we may obtain the power series expansion for the projection
of the section Cy in the plane x ^ - 0 , namely,
(4.2)

x^/xx = l|_c^w(x2/x1 )2 y B 1 (x2/x:L)3 ^ eeeJ e

The equations of any four-point conic of the section Cy at
the point y are found, from equation (4 *2 ), to be (4 .1 ) and
(4*3)

~ c*x22/2 /SA- 2^6Bix2x^/c’ y k x ^2 = 0 ,

where k is a parameter.

The polar line of the point z

with respect to this conic is given by equations (4 .1 ) and
(4.4)

c,2X2+2/3 2B 1 X i ^ = 0 .

Elimination of A between equations (4.1), (4.4) shows that
as the plane rc revolves about the line yz, the locus of
this polar line is a plane through the line z £ :
(4.5)

c

*2 x 2 + 2

Similarly, as the plane n revolves about the line yz,
the locus of the polar line of the point y with respect to
any four-point conic of the section C 2 at the point z is a
plane through the line y 7 :
(4 .6 )

c2x1 y 2 y2B ^ x ^ = 0 .

Thus we reach the following conclusion:
As the plane re revolves about the line yz, the locus

- 20 -

of the principal point associated with the points y, z of
the plane sections Cy, Cz describes a line

jL

, whose equations

are (4.5), (4.6)
This line 1 intersects the lines
=

= 0 in two points*

=0 and

If we choose these two points

respectively for the points y\ and ^ , then
(4.7)

Bx -0,

B* = 0;

that is,
(4 .8 )

p = c( ry/ r + 3a - <T),

q* = o' ( (ij(b + 3b* - ^ ).

Hereafter it will be supposed that the differential
equations (1 *1 ) are in the canonical form for which the
conditions (4*7) or (4.6) are satisfied.

Accordingly, by

means of equations (1 *3 ), (2.5), (4.6) the power series
expansions (3 *6 ), (3 .1 1 ) for the surfaces Sy, S 2 in the
neighborhood of the points y, z become, respectively,

(4.9)

y^/yi = o,(y2/yi)2/2/9+q(y3/yi)2/2o2
- c’ltyg/yj^)2 (73 /73^)/2c/6 - qA*(y2/yi) (73 /7 1 )
+ qM(y3/y1)V6o^ - c’H y g / y ^ V s /S3

...

+ q[Mv +M2 +(P - <£v)m+ 3L ‘J (y3/y1)V24ci»-t...,
(4.10)

Z$/Z2 =c(z 1 /z2 )2/2y +p'(z^/z2 )2/2c*2
-cl’(z-y/z2)2(z^/z2 )/2c *V-p*A(zi/z2)(z^/z2)2/2c’2/
+p»M’(z^/z2 )3/6o*3 - cL* (z1/z2 ) V 8
+ p*

+

+{p*

V^ +

...

- - f u )M * + 3 L ] ( z 4 / z 2 ) V 2 4 o * ^

"h ♦*•
Irom equations (1 *1 ), it follows immediately that the

- 21 -

point 5 is, in the osculating plane of the v-curve of the
net Ny at the point y if, and only if, n =0*

A similar

argument holds for the u-curve of the net Nz at the point
z.

Moreover, from equation (2*10), the point y^ coincides

with the point v\ if, and only if, I -0; and similarly,
the point z_^ coincides with the point £ in case 1 1 = 0 •
Buzano [3] and Bompiani T2] have shown the existence
of a projective invariant, together with metric and pro­
jective characterizations, determined by the neighborhood
of the second order of two surfaces (T-, (r* at two ordinary
points 0 , 0 * in ordinary space under the conditions that
the tangent planes of the surfaces (T, (T* at the points
0 , 0 * be distinct and have 00 * for the common line.

For

the focal surfaces Sy, Sz at the points y, z, this invariant
is easily found, from Bompiani*s note [2] and expansions
(4 .9 ), (4 .1 0 ), to be $ /(16k).
As the plane 7r(4*l) revolves about the line yz the
locus of the principal conic having four-point contact
with the section C

at the point y is a quadric cone with

vertex at the point ^ » whose equation is found, by
means of equations (4 *1 ), (4 .3 ) and the conditions (4 *7 ),
to be
(4*11)

2 /3 x1x4 “ c*x22 = 0 .

Similarly, the locus of the principal conic having
four-point contact with the section Cz at the point z is

- 22 -

a quadric cone with vertex at the point ^ :
(4*12)

2 V x 2^

- cx^^O.

From equations (2.5), (2.10), (2.11), we find the
equations of the line y^z.i to be
(4 *1 3 )

cx^-r 1x ^ = 0 ,

c’Xgtl'x^^O,

which intersects the plane (4 *1 ) in a point with the
coordinates
(4.14)

(-I/o, -1 1A /c *, 1 , A).

If a four-point conic (4*3) at the point y of the section
Cy of the surface Sy with the plane (4 .1 ) passes through
the point (4.14), then we can determine, by observing the
condition B^ = 0 , the parameter k in equation (4 *3 )•
(4.15)

k = 1 / c + X ,2 X / ( 2 c ' /3 ),

and the equation, other than (4 .1 ), of the conic becomes
(4 .1 6 )

-(c */2/3A)x22 + (I/c -M* 2 X / 2cfp )x^2 = 0 .

As the plane (4.1) revolves about the line yz, the locus
of this conic is a quadric cone with vertex at the point
y^, whose equation is found, by eliminating A from
equations (4 .1 ), (4 .1 6 ), to be
(4.17)

*3 X4

-(o,/2/3)x2

2

+tt/o)x3 X 4 + £ ' 2 /2c’/3)

- 23 -

Similarly, we may obtain the quadric cone with vertex
at the point
(4.18)

x2x^ -(c/2h)x^2+(l2/2cK)x^2+(l’/o’ )x 33:^ = 0 .

The line

intersects the quadric cones (4*17), (4.18)

in the points Yj , 5 and two other points with the
coordinates
(4.19)

(0 ,0 , cl»2, -2c'/3l),

(4.20)

(0, 0, 2cri*, -c’l2).

The two points (4.19), (4.20) coincide in neither y) nor 5
if, and only if,
(4.21)

II* - 4K,

and they are separated harmonically by the points 77 , ^
in case
(4.22)

II* + 4K=0.

5 . Moutard Q.uadrics of the Surfaces Sy , S 2

It is known that for a nonasymptotic tangent at an
ordinary point of a surface there is a quadric of Moutard,
which is the locus of the osculating conic at the point of
the plane section of the surface made by a variable plane
through the tangent.

In this section we shall find the

- 24 -

equations of the quadrics of Moutard for the tangents yz,
y>7 of the surface Sy and also for the tangents zy, z£

of

the surface S2 *
By means of equation (4*1) and the series (4*9), we
may obtain the expansion for the projection in the plane
x^— 0 of the section Cy of the surface Sy made by a
general plane (4*1):
(5*1)

Xo/x-i -

c» /Xo\2

c f /cfq -

c 'IX

_

The osculating conic of the section Cy at the point y is
given by equations (4*1) and
(5*2)

x-|X0 X ^

o* Xq ^ -

1 /c’q _ cfIX _ LX* ^x-,2 = 0*
0
2P )

As the plane (4*1) revolves about the line yz, the locus
of this osculating conic is the quadric Qy^u ^ of Moutard
for the tangent yz of the surface Sy, whose equation is
found, by eliminating X

(5.3)

from equations (4*1), (5*2), to be

x,x, - _c_' Xg2 - _g_
2/3

x 32+ I. x -Xjl +J;_ xji.2 =

2c2

0

3

0.

2 c V>

Similarly, the equation of the quadric Qz'v ^ £f Moutard
for the tangent yz of the surface S 2 is

(5*4)

XpXo **—o_ x^2 _l —Ly Xo^-f
d ^

2r

2cr

*—2——-x42 — 0 •
cf

2c1

- 25 -

In order to find the equation of the quadric Q y ^

of

Moutard for the v-tangent of the surface Sy at the point y,
let us consider a general plane through the line y

:

(5*5)
By means of equation (5*5) and the series (4*9)» we may
obtain the expansion for the projection in the plane
of the section
(5.6)

of the surface Sy made by the plane (5 *5 )*

=
3^

x^= 0

2

c

6

* ___ f
kc^yU3 ( 2 /3

c}u.'cs^}

p

+ ^5[ m t + m 2 +( p -

The osculating conic of the section Ty, at the point y is
given by equations (5 *5 ) and
(5.7)

ZiXo - _ L i A L l ± *

%al2j3

+(P - V

(3

+ 4 ? K
6

- m2/3

M + 3l ’J} x 22 - ^ * 2 x3 - i i 7 x32 = 0 ‘

Elimination of ^ from equations (5*5), (5*7) gives the
equation of the quadric Q y ^

of Moutard for the v-tangent

of the surface Sy at the point y:
(5.8)

- c ’x 22/2/3 - <jx32/2c2 +A*x2xJ!f//3 - M x 3x ^/3c
+ _ 1 _ [ M 2 - 3M, - 3(P- ^t )M -9L*J x ^2 =0.

Similarly, the equation of the quadric Q_'u ' ££ Moutard

- 26 -

for the u-tangent of the surface Sz at the point z is,
(5.9)

X2X3 - ^ 7 x12 +A XlX3+_ l _ [ M ' 2 - 3M*U - 3(P’

- y u)M* - 9LJ x32 -ML. x3x4 - _ | ! ^ x 42 = 0 .

It is obvious that if the quadrics Q y ^ ,

pass

through the points ^

then the focal nets N , N z are
¥
restricted, respectively, by the conditions 1 = 0 , L f= 0 .
The polar planes of the points 7 , 5
quadric Q y ^

with respect to the

and with respect to the quadric Qz^

are,

respectively,
(5*10)

qx3 - clx^ = 0,

(5 .11)

+ Ix3/c t Lx^/c f/3 = 0,

(5.12)

x2 + L*x3/cr +I»x4/c* =0,

(5.13)

c»I*x3 - P fx4 = 0.

If the planes (5*10), (5*13) pass through the points 5 >7
then 1 = 0, I ,= 0, respectively,

Furthermore, these two

planes coincide in case the focal nets Ny, N2 are restricted
by the condition
(5.14)
The line 7 5

H f= § •
intersects the two planes (5.11), (5.12) in

two points with coordinates

- 27 -

(5.15)

(0 , 0 , cL, - c'/3I),

(0 , 0 , cri», - c'L*).

The points (5.15) coincide in neither ^ nor 5 if, and
only if,
(5.16)

LL» = II *K,

and they are separated harmonically by the points >7 , 5

in

case
(5.17)

LL» + II*K = 0.

On the other hand, the polar planes of the points z,
7 ,%

with respect to the quadric Q y ^

have the equations,

respectively,
(5.IS)

°'x2 " a ’3:4 ==0>

(5.19)

3qx3+ c M x ^ O ,

(5.20)

Xj'+A’ x 2

-_M_ x 3 +_1[M2 -3MT-3(P-^t )m-9L*]x4 =0.

If the planes (5*16), (5.19) pass through the point
then n=0, M = 0 , respectively; and a similar argument can
made with regard to the quadric Q,_^u ^. Moreover, the plane
(5 .1 9 ) coincides with the polar plane of the point 5 with
respect to the quadric Qz^
(5.21)

in case

MM* - 9 § .

The line y ?] intersects the planes (5.11), (5.20) in

- 28 -

two points with coordinates
(5.22)

(X, 0, -c, 0),

(M, 0, 3c, 0).

These two points coincide in neither y nor ^

if, and

only if,
(5.23)

31+ M =0.

Further, if the points (5 .2 2 ) are separated harmonically
by the points y, y\
(5.24)

then

31— M.

Likewise, we can discuss the conditions similar to (5.23),
(5.24).
Finally, if the planes (5.10), (5.13) coincide, and
if the points (4 *1 9 ), (4 -2 0 ) are coincident or separated
harmonically by the points
N

z

\ , then the focal nets Ny ,

are restricted by the conditions (5.14) and

(5.25)

§ = ± 4 K.

Now we are in a position to determine geometrically
the unit point of the coordinate system.

To this end, we

observe that the locus of a moving point, whose polar planes
with respect to the quadrics Qy^v ), Q,z^

intersect the

generator yz at the same point, is a quadric, its equation
can easily be found to be

- 29 -

(5.26)

cc,x1 x 2 - cA’x ^ ^ - c’Ax^x^ + (AAf

= 0.

This quadric cuts the cubic curve of intersection, besides
the line 7
points.

, of the two cones (4 .1 1 ), (4 .1 2 ) in two

It is easily seen that i^f m^ and n not van*ah at

the same time for the congruence yz. then we may take one
of these two points as the unit point of the coordinate
system, and therefore we have
(5*27)

c = 2 r,

c*=2/3,

3p*q t Im'n = 4 (m,q +np*)•

6 . Segre-Darboux Nets on the Surfaces Sy, Sz

It is known that associated with an ordinary point 0
of a surface S there are three-parameter family of quadrics
each of which has second order contact with the surface S
at the point 0.

A general quadric of this family intersects

the surface S in a curve with a triple point at 0.

In

particular, if the three triple-point tangents of the curve
of intersection coincide, then the quadric is called a
quadric of Darboux, and the corresponding coincident triple­
point tangents and the curve of intersection are respectively
called a tangent and a curve of Darboux at the point 0 of
the surface S.
In order to find the equations of the configurations
mentioned above for the surface Sy at the point y, we first
consider the quadrics having second order contact with the

- 30 -

surface Sy at the point y.

The equation of a general one

of these qudrics is obtained by writing the equation of
the most general nonsingular quadric and demanding that
the series (3 »4 ) satisfy this equation identically in au,
A y as far as the terms of the second degree#

The result

can be written in the form
(6 #1 )

^i^ -

cy x 22 2 ^3

q

x ^2 + (k2x2 + k^x^ + k ^ x ^ x ^ 0 ,

2c

where k2 , k^, k^ are parameters.

Each quadric (6.1) cuts

the surface Sy in a curve with a triple point at y, whose
tangents are in the directions satisfying the equation
(6*2)

3cf/32k2du3 -h3cf/3(ck^ - I)du2dv + 3q(/3k2 - A*)dudv2
f q( 3ck3 +M)dv^ = 0 .

It is not difficult to verify that if the binary cubic form
that appears in equations (6 .2 ) is a perfect cube of a
linear form, then
(6.3)

k2 =A*/4/3,

k3 =(I - M)/4c,

and therefore the equations of the curves and the pencil
of quadrics of Darboux at the point y of the surface Sy
are, respectively,
(6.4)

3c*ySA*du3 - 3o*/3(3I tM)du2dv -9qA*dudv2+q(3I+M)dv3 = o,

(6 .5 )

4 c2/3x^x^-2 c2 c 1x2 2-2 q/3X 32 1c2A 1X2X4 +c^5(I -M)x ^xj^-f-k^x^2 =0 •

- 31 -

Similarly, the curves and the pencil of quadrics of
Darboux at the point z of the surface Sz are respectively
given by the equations
(6.6)

p* (3I,tM* )du3-9ptAdu2dv-3cr(3I*tM* )dudv2+3crAdv3 = o,

(6.7)

2c *2cx^2-e1^Ax^X3“4c *2/X2X3~c *Y{1 1-Mf)x^x^-f^p V x ^ 2
•fIc^x^2 = 0,

where k^ is a parameter.
The equations in local coordinates of the tangents of
Darboux at the point y, z of the surfaces Sy , Sz may easily
be obtained from equations (6 .4 ), (6.6), (7.4), (7.5); the
result we shall omit here.
The polar line of the line y£

with respect to the

pencil (6.5) of quadrics of Darboux at the point y of the
surface S_ is
y
(6*8)

x ^ - 4c/3X^ + cA ,X2 + /3(I - M)x3 =0,

which intersects the line yz in a point
(6.9)

(-A*, 4/3, 0, 0),

and passes through the point z or 7 in case n = 0 or I = M.
Similarly, the polar line of the line Z 7 with respect to
the pencil (6.7) of quadrics of Darboux at the point z of
the surface Sz is
(6.10)

x 3 = c ‘Ax1 + 4c,r:ic2 + r(l' -M')x^ = 0,

- 32 -

which intersects the line yz in a point
(6.11)

( 4 X , - A, 0, 0),

and passes through the point y or 5 in case m* = 0 or I* = M f•
Moreover, if the two points (6.9), (6.11) are coincident
or separated harmonically by the points y, z, then the
focal nets Ny, N z are restricted by the conditions
(6.12)

AA* =■ ± 16K.

From equations (6.4), (6.6)

it is easily seenthat

the curves of Darboux correspond

on the

two surfaces Sy ,

Sz if, and only if,
(6.13)

W (u)=0,

(3I+M)(3I,+ M») = 9AA*.

It is known [4, p. 283] that if the curves of Darboux
correspond on the two focal surfaces of a congruence, the
congruence is a W congruence and both surfaces have the
property of being isothermally asymptotic.

Thus the

conditions for the both surfaces Sy, S2 to be isothermally
asymptotic at the same time can be reduced to the form (6.13)•
On the other hand, a necessary
for the u-curves or the v-curves

and

of the

curves of Darboux is n = 0 or (5.23).

sufficientcondition
surface Syto be

It is known that if

the curves of one family of a conjugate net are curves of
Darboux, then the curves of the other family must be the
corresponding curves of Segre.

Such a net is called a

Segre-Darboux net. Combining the above results and the

- 33 -

ones in

§§ 4, 5 we obtain the following theorem:

The focal net

in a Segre-Darboux net with the u-

curves as curves of Darboux if, and only if, the -point $
is in the osculating -plane of the v-curve of the netNy at
the point y*

The focal net N^. in a Segre-Darboux net with

the v-curves as curves of Darboux if, and only if, the
line y y intersects, in the same point, the polar planes
of the point

$ with respect to the quadrics Qy(u ^,

of Moutard at the point y of the surface S •
J
We shall call a congruence a Segre-Darboux congruence
when the two focal nets of a congruence both are SegreDarboux nets*

Noticing the theroem concerning the other

focal net N z and similar to the above one, we may obtain
a geometric interpretation for a Segre-Darboux congruence
yz*

7* W Congruences

The differential equation of the asymptotic curves
on the surface Sy is
(7.1)

Ldu2 + 2Mdudv + Ndv2 = 0,

where the coefficients L, M, N are the determinants of the
fourth order defined by
(7 .2 )

t = (yuu,y,yu ,yT )»

M=(yuv.y>yu *y-v>>

H = (yw»y»yu*yv> •

- 34 -

By means of equations (1.1), (3*1), it is easy to write
equation (7*1) in the form
(7*3)

c'/3 du2 -f qdv2 = 0.

The equations, in local coordinates, of the asymptotic
tangents to the surface Sy at the point y may easily he
obtained from the expansion (4*9) of the surface Sy, or
else from equation (7*3) and the fact that the tangent at
the point y to a curve C x

belonging to the family defined

on the surface Sy by the differential equation
(7*4)

dv - X du = 0,

where X

is a function of u, v, has the equations in local

coordinates
(7-5)

= p x3 - e a x 2 =0.
Similarly, the differential equation of the asymptotic

curves on the surface Sz is
(7*6)

p*du2 -f- c X d v 2 -0.
The asymptotic curves on the focal surfaces Sy, Sz

correspond in case equations (7*3), (7.6) are equivalent.
Then the u-tangents of the net N

or the v-tangents of the
V

net N

form a congruence of the special type called a W
z
congruence. Thus we reach the result that the u-tangents

congruence in c_a.se the Weingarten invariant W f„\ or W! .
1 'u/
(v)
defined by equations (2*5) vanishes*
From equation (2.10), (l.l), (1.2), (1.3) by differ­
entiation and substitution, any derivative of y-^ can be
expressed as a linear combination of y, z, ^ , 5 , In
particular, one obtains
/

y

— (H - cf d / p )y H-c

7 ,

yiv=[°S - *c?//3-(ct/p)v]y + nz +(o8+ ov.o2f//5)7 +q5,

(7.7)

y.
= (*)y + 3 Hz + (*) 7 ,
< luu
1
yiUV

+

ylvv^(*)y

• n fiv/t3 + np/°

+ (*)7

+q(p/c - c^/c’ - /3v/^3 -t qv/q) 5 .
From equations (7.7) and the ones similar to (7.1), (7.2),
the differential equation of the asymptotic curves on the
surface Sv

sustaining the first Laplace transformed net

N,, of the net Nv is found to be
y
y±
(7.S)

c '|6 Hdu2 + qftdv^ = 0.

Thus the v-tangents of the net Ny form a W congruence in
case the Weingarten invariant
(2.5)

vanishes.

defined by equations

- 36 -

Similarly, the u-tangents of the net N z form a W
congruence if, and only if, the Weingarten invariant
vanishes.
From the relation between the Weingarten and the
Laplace-Darboux invariants of a conjugate net, we know
immediately that

, ft, and

are respectively the

tangential invariants of the focal nets N

and Nz .

Now we proceed to give a simple geometric interpre­
tation for the condition for the u-tangents or the vtangents of the net N

to form a W congruence.

The equation of any quadric having the lines yz, y^ ,
z £ as generators can be written in the form
(7.9)

x1x/f-fk1x2x3 + lc2x3x^ = 0,

where k., k2 are parameters.

A general quadric (7.9)

intersects the surface Sy in a curve with a double point
at y, whose tangents are found, from the series (4*9), to
be
(7.10)

x ={c *//3)*22 + 2^

12X 3 + qx32/c2 = 0.

If these two tangents coincide, then
(7.11)

= c'q/c2/3 .

Similarly, the quadric (7*9) cuts the surface Sz in a curve
with a cusp at z if, and only if,
(7*12)

c*2 //cpf.

- 37 -

The conditions (7*11), (7*12) hold simultaneously in case
W(u j= 0.

Thus we arrive at the following theorem:

A necessary and sufficient condition for the congruence
yz to be a W congruence is that there exists a quadric
(and therefore one-parameter family of such quadrics),
which has yz, y ^ , z 5 as generators and whose curves of
intersection with the focal surfaces Sy, Sz have cusps at
the points y, z respectively.
Similar statements can be obtained for both the
congruences yrj and z 5 #
A conjugate net whose Weingarten invariants both
vanish is called [9 , p. 1077] an R net, each family of
curves of an R net has tangents that form a W congruence.
From the above theorem, we may also interpret geometrically
a congruence when either of its focal nets or both are R
nets.

8 . Curves of the Focal Nets Ny, N z

By putting a v to zero, expansions (3.5) become the
nonhomogeneous local coordinates of a point Y near the
point y and on the u-curve through y of the focal net Ny ,
and therefrom we may easily obtain the power series expan­
sions for the u-curve of the focal net Ny in the neighbor­
hood of the point y, namely,

- 38 -

y3 _

»■
6

(8 .1 ) <
i t -

yx

...

p

W

° ’

Tp

24 /B3
12
71 J

Ui/

c*L (72^^
8 p

+

^yl

Similarly, by putting a u to zero in expansions (3.5), we
reach the following power series expansions for the v-curve
of the focal net Ny in the neighborhood of the point y:
n pr3\2
n (M+ ft/A* )/y3\3
La.
yi 2c2 1^1/ 6c3

n

[Mv f■M'

24c'

+ 3L* +( ^+211 ft)/A* + (P- 4>y )(M t ^ / A f)J^y3j4

(8 .2 ) <
-f
1 •

y-!

• # j

2 c2 'yx/

y 3^3
q [Mv+M2t(P- ^>V )M
-J----6c3 \ylI
24c^

+ 3L ' J ( i 2] 4 + . . . .

Analogous expansions for the u-, v-curves at the point
z of the focal net N„ can be written by making the substitutions (3*8), (3.9) on the symbols.
Prom expansions (8.1), (8.2), the quadrics (6.1)
having second order contact with the surface Sy at the
point y are the quadrics having third order contact with

- 39 -

both the u-, v-curves of the surface Sy at the point y
in case
(8.3)

k2 = 0 »

k3 = - M/3c.

Thus we find that any quadric of the -pencil
(8.4)

^i3^

- qz^2/2 c2 -MX3X4 / 3C

= 0,

where k^ Is arbitrary, has third order contact with the
parametric curves of the surface Sy at the point y.

If a

unique quadric of this pencil is desired, we may choose
the one that passes through the covariant point £ • The
polar plane of the point

with respect to any quadric

of the pencil (8 .4 ) is the plane (5 *1 9 )•
Among the quadrics (6.1) there is a pencil having
fourth order contact with the u-curve (8 .1 ) of the surface
S

y

at the point y.

(8.5)
with

k2 = 0 *
arbitrary.

For this pencil we find
k4 =L/2c73
The quadric Qy

,
(5*3) of Moutard for

the u-tangent of the surface Sy is a unique quadric of
this pencil [8 , p. 698 ], and for this quadric we have
k-j =• 1/c •
Among the quadrics (6.1) there is also a pencil having
fourth order contact with the v-curve (8.2) of the surface
Sv at the point y.
J

For this pencil we find

-

*

3

ko -

= - M/30,

(8.6) J
k4 = I|^[M2 - 3Mr - 3(P -^V )M - gL» + 9nA»//3 - 18nk2J .

9 • Correspondences Associated with the Focal Nets Ny , N z

Let us consider a curve Cx

passing through the point

y and belonging to the family (7 *4 ) defined on the focal
surface S ; and let the parametric representation of the

Jf

curve Cx

be

u = u(w),

(9.1)

v = v(w).

The parametric u-, v-tangents at points of the curve CA
generate two non-developable ruled surfaces Ry(u ), Ry ^v ^
respectively.

The points

T = ty - z,

(9.2)

T = ty -

7

,

where t, t are functions of u, v, lie on the parametric u-,
v-tangents through the point y respectively, and the line
determined by them lies in the tangent plane

0

of

the focal surface Sv at the point y.
j

From system (1.1) and equations (1.2), one easily obtains
Tu = (tu + t c O y +(t/3- b')z - o*5 ,

(9.3)

i

Tv = (tv +at - r ) y + c t

"7

- <^z,

- 41 -

Tu =(tu + t<^- e)y+(t/3- f)z - g rj,
Tv = (tv -f-at - r)y - nz/c + (ct - s)^7 - <l$/c.
The tangent planes to the ruled surface Ry ^
T and to the ruled surface R y ^
in a line

at the point T intersect

which passes through the point y.

be a point on the line

at the point

There must

given by an expression of the

form
X = 's + kj_z

.

The plane of the points y, X, T is to be tangent to the
ruled surface Ry^u ^ at T, and therefore must contain the
tangent to the curve traced by T as the point y moves
along the curve Cx • It is clear from the first two of
equations (9*3) that a point on this tangent

is given by

Tw = (*)y + (*)z+ctvw ^ - c’u ^ ;
that is to lie in the plane yXT, which can occur if and
only if ^2 = - ctx/c*.

Similarly, the plane yXT is tangent

to the ruled surface R y ^
-f - n A / c ) / q x *
(2 )

at T if and only if

= -c (t/3

Thus we obtain a correspondence(2)

This correspondence has been introduced by the

author [6, pp. 3S1-3S2J.

The line A (9*5) is the analogue

for a conjugate parametric net of the R fx "associate of the
reciprocal of Jt2 (9*4) defined by Bell [1, pp. 390-391J.

- 42 -

between the line iz in the plane x^-0:
(9.4)
and the line

through the point y:

Xj-tct X x^/c * = 0,

(9.3)
x2 + c(t/3 - f - n x/c)i^/qx = 0.
Conversely, if equations (9.5) are written in the form
(9.6)

x 2M

=

= x2/<3 ,

then
f t = - c Tift/c x ,
(9.7)

J>t=- q\Hl /c|3 + (f + nx/c) d / p
f = <5

*

'When the line (9.4) passes through a fixed point
(xj, x2>, x|, 0), the corresponding line (9.5) describes a
plane, its equation is
(9.8)

jJLL x^x2 +

x 2x 3 +

[X1 + (f + n X/c)x|/^J

=

Consequently, a point (xf, x|, x*, 0) in the plane x ^ = 0
and the plane (0, u2 , u^, u^) through the point y are in
the following correspondence:

- 43 -

r u2 - - qA/(C|3) x*,

(9.9)

J <ru3 == - c V ( c x ) x*,
0“u ^ = x * + i,(f + n x / c ) x | ,

(<T ^ 0).

This correspondence becomes a polarity with respect to a
quadric (and therefore ooL quadrics) if and only if x
satisfies

(9.10)

x 2 = c » /3/q,

that is, if and only if the tangent of the curve Cx

at

the point y is an associate conjugate tangent of the
parametric conjugate net Ny.

The equation of any one of

these quadrics is then of the form
(9.11)

c/D x - ^ - c* x2x3 + (A* t I/B) x3x^-f-kj^x^2 = 0 ,

where k^ is a parameter.
In a similar manner, with regard to the focal net N_
and the focal surface S 2 we may obtain the polarity with
respect to the following pencil of quadrics

(9.12)

c f/ D ’ x2x3 - c x-^x^ -f{A + 1 9/IT*) x ^ f k ^ 2 = 0,

where k^ is a parameter.
The polar planes of any point (x*, x2j, 0, 0) except
the points y, z on the line yz with respect to any quadrics
(9.11), (9*12) are respectively

- 44 -

C * X $ X o - o / D X?X, = 0 ,

(9.13)

)
jL3
1 4
I cf/D* z|x 3 " cxix4 ==i0>

which coincide if and only if K =■ § . Thus we obtain
another geometric interpretation of a W congruence:
A necessary and sufficient condition for the congruence
yz to be a W congruence is that the "polar planes of any
point. except the points y, z, on the generator yz with
respect to the two pencils (9*11), (9*12) of quadrics be
coincident*

10. Axis Congruences and Ray Congruences

The axis at the point y of the net Ny is defined to
be the line of intersection of the osculating planes of
the u-, v-curves of the net Ny at the point y.

From

equations (1.1), (3.1) it is easy to obtain the equations
of this axis, namely,
(10.1)

x^ = qx2 - nx^ = 0.

The point Y defined by
(10.2)

Y = ky + nz +q 5

is on the axis.

(k scalar)

Vi/hen the point y varies along a curve Cx

of the family (7.4) on the surface Sy, the point Y
generates a curve whose tangent at Y is determined by Y
and Y f, which can be expressed as a linear combination of
y, z, r( , *5

by using equations (1.1), (1.2), (1.3), namely,

- 45 -

(10.3)

Y ^ f k * t ^ k + m V c ’ +(ak-f-nr + e fq)Aj y

-hqs* + (n^. - m i S + f*q)A] z

+ [/3k -fb’n

+ ( c k X + p V c * ) ^ + [c’n + qu + qrf + (qv + q<f

-04/0MAJ 5

.

In the expression (10.3) equating the coefficient of Y ‘to
zero and the ratio of the coefficients of z and $ to n/q
and making use of equations (2.5), we obtain two conditions
on the functions k and X which are necessary and suffi­
cient that the axis may generate a developable surface and
have Y for focal point when the point Y variesalong

the

curve Cj^ ; these two conditions are
(10.4)

§+kX^0,

c ,/3N + c ,/3 k + q ^ A = 0.

Elimination of _x from these two equations gives

(10.5)

0*^6 k2 + 0*16 Nlc - q^ft = 0.

If k^ and k2 are the two roots of this equationregarded
as a quadratic equation in k, the two points Yj_ and Y 2
defined by the formula (10.2) with k^ and k2 , respectively,
in place of k are the focal points of the axis, and the
loci of these two focal points when u and v vary are the
focal surfaces of the axis congruence of the net Ny . On
the other hand, on eliminating k from equations (10.4 ) and
replacing x by dv/du, we obtain the differential equation
of the axis curves of the net Ny , in which the developables

—

of the axis congruence
(10.6)

46

—

intersect the surface Sy ,namely,

c*^6 § du2 - c’^Ndudv - qftdv2 = 0.

The tangents of these axis curves

are called the axis

tangents of the net N .
«y
Since at each point of a harmonic conjugate net the
axis tangents separate the tangents of the net harmonically
[13, P* 215 J, it follows immediately that the net Ny is
harmonic in case N = 0.

Similarly, the net N z ijs harmonic

if, and only if, N*=0.

From equations (7.3), (10.6) we

may also verify the result fl2, p. 319] that the axis
curves of the nets Ny, Nz form conjugate nets in case
§ = ft ,

respectively.

We shall next determine the developables and focal
surfaces of the ray congruence of the net Ny.

The point

X defined by
(10.7)

X^y^+kz

(k scalar)

is on the ray of the net N

corresponding to the point y.
V

When the point y varies along a curve of the family (7.4)
on the surface S , the point X generates a curve whose
tangent at X is determined by X and the point X*, which
can be expressed as a linear combination of y, z, yj ,5
by means of equations (1.1), (1.2), (1.3), (2.5), namely,
(10.8)

X*= |H - ctJL/{3 + [a^ - <TT ta2 - a S'- (log^)^

- 47 -

- a{log/3 )T +cr+^3r k j x | y + [ c u + og+o(a - <P
+ 3

+ C T/c - ^3y /(i )A] Z + C [<A + (cv /c -

ts)x]iy + ( q A + c'k);

Ot/j3

(X’=d2:/du, ...)

In the expression (10.8) equating the coefficient of $ to
zero and the ratio of the coefficients of y and ^

to -f//3

and making use of equations (1.3), we obtain two conditions
on the functions k and X

which are necessary and sufficient

that the ray may generate a developable surface and have
X for focal point when the point y varies along the curve
Cx ; these two conditions are
e #k -f q X = 0,

(10.9)
|9H + [ p r k - e(fv t f<f- r/3 - fs) ] A = 0
From the second and the fourth of the integrability
conditions (1.4) and equations (2.5), we may further reduce
the two conditions (10.9) to
(10.10)

e * k + q A = 0,

Elimination of X
(10.11)

H + ( r k - N ) X = 0.

from these two equations gives
c » r k 2 - c TNk - qH = 0.

If k. , k~ are the two roots of

this equation, then the

corresponding points X^, Xg are the focal points of the
ray, and the

loci of these two focal points when u andv

vary are the

focal surfaces of the ray congruence of the

-

kB

-

net Ny . On the other hand, on eliminating k from equations
(10*10) and replacing X

by dv/du, we obtain the differen­

tial equation of the ray curves of the net Ny, in which
the developables of the ray congruence intersect the
surface Sy, namely,
(10*12)

c'Hdu2 - c’Ndudv - q r d v 2 = 0.

The tangents of these ray curves, called the ray tangents
of the net N , separate the tangents of the net EL harmoni-

y
j
cally in case N = 0, that is, in case the net is harmonic

[13] * The ray curves of the net EF form a conjugate net
Jr
r12] in case E = K.
In a similar way, we can discuss the ray congruence
of the net N z .

11* The Congruences z^, y$ and the Principal Congruence rj^

As the point y varies on the surface Sy, the line z ^
describes a congruence.
(11*1)

P — ^ + kz

The point P defined by
(k

scalar)

is on the line z ^ . When the point y varies along a curve
of the family (7.4) on the surface Sy, the point P generates
a curve whose tangent at P is determined by P and the point
P f, which can be expressed as a linear combination of y, z,
^9

by means of equations (1.1), (1*2), namely,

- 49 -

(11*2)

P ’= (e + r x + r k A ) y + (f + nX/c-f-bfk + <fkX
t k T)z +(g + sa) y\ + (qAtcc’k) $ /c
(P* = dP/au, ...)*

In the expression (11*2) equating the coefficients of y, 5
to zero, we obtain two conditions on the functions k and A
which are necessary and sufficient that the line z^ may
generate a developable surface and have P for focal point
when the point y varies along the curve Cx ; these two
conditions are
(11.3)

cc’k f q x = 0,

J +(c X k + L 1 )A = 0.

Elimination of A from these two
(1 1 .4 )

equationsgives

e^'r^-t-ee'L'k - q J = 0 .

If k-^, k2 are the two roots of this equation, then the
corresponding points P^, P2 are

the focalpoints of the

line z 7 , and the loci of these

two focalpoints when u

and v vary are the focal surfaces of the congruence z y) of
the net Ny * On the other hand, on eliminating k from
equations (11*3) and replacing

A

by dv/du, we obtain the

differential equation of the curves in which the developables of the congruence z

of the net

surface Sy , namely,
(11*5)

c TIdu^ -+c ’I^dudv - q r d v 2 = 0.

intersect the

- 50 -

The tangents of the curves (11.5 ) separate the tangents
of the net N

harmonically in case L* = 0*

Moreover, the

curves (11.5) form a conjugate net if. and only if, J=K.
Finally, from equations (2.7), (2.10) it follows that if
the point y^ coincides with the point ^ and if the eurves
(11.5) form a conjugate net.then the conjugate net Ny has
equal Laplace-Darboux invariants.
Similarly, we can discuss the congruence y£ .
Finally, we shall call the congruence
cipal congruence of the congruence yz.

the prin­

The point Q, defined

by
(11.6)

Q =

-f- k 'c,

(k

scalar)

is on the line ^ $ • When the point y varies along a curve
of the family (7.4) on the surface Sy, the point Q, generates
a curve whose tangent at Q is determined by Q and the point
Q* given by
(11.7)

Q,* = (e + r x + m ’k/c’t e ’kxly + lf t n x / c + s fk
tf fk\)z t(g t s x -f-p’k/c1)rj + (gx/c -f-rfk
-fg^xtk*) 5 ♦

In this expression equating the coefficients of y, z to
zero, we obtain two conditions on the functions k and x
which are necessary and sufficient that the line -75 may
generate a developable surface and have Q, for focal point
when the point y varies along the curve

; these two

-

51

-

conditions are
(11.8)

e + r x + i ’k/c* + e*kx=0,

f + nx/c + s'k + f *kX= 0.

Elimination of X from these two equations gives
(11.9)

c2m* (B*I *L - GJ* )k^ + cc ’G(II ’K +IX* - JJ*
-G)k + c»2n(BIL -

If k]_, kg

= 0.

are the two roots of this equation,then the

corresponding points
line

OT)

, G;2 are the focal points of the

5 , and the loci of these two focal points when u

and v vary are the focal surfaces of the principal congru­
ence ^ 5*

On the other hand, elimination of k from equations

(11*9) and replacement of X by dv/du yield the differential
equation
(11.10)

(BI - JL)du2 + (IX'K - LL’ - JJ*+G)du dv
+ (B*X' - J'l1)dv2 =0.

The developables of the principal congruence ^5 intersect
the surfaces Sy, Sz in the curves respectively represented
on Sy, S2 by this differential equation.

We shall call

these curves the principal curves of the nets N , Nz . The
tangents of these principal curves will be called the
principal

tangents of the nets Ny,

The principal tangents of

one

of the same net harmonically in case

N z.
net separatethe tangents

- 52 -

(11.11)

G+II'K = JJ* + LL*.

Moreover, necessary and sufficient conditions for the
principal curves to form conjugate nets on the surfaces
Sy, Sz are, respectively,
(11.12)
(11.13)

ft(BI - JL) + DH(B*I * - J»Lf)=0,
D*K'(BI - JL) +

- J»L»)=0.

Finally, the principal curves form conjugate nets on the
both surfaces Sy , Sz at the same time if, and only if,
(11.14)

HKK* = § ft§».

12. Osculating Linear Complexes and
Associated Linear Complexes

In this section we shall find the equations of the
osculating linear complex along a generator of a W congruence
and the associated linear complexes of the focal points of
a generator of a congruence(3)„
First of all, we introduce line coordinates.

In the

customary way, the pltickerian homogeneous coordinates of
a line joining two points with coordinates (y^,
(z^,

z^) are defined to be the six numbers
(3) Of. [1 0 , p. 167 J;

[7 , pp. 300-303 ].

y^J,
» ^13»

- 53 -

(12 .1 )

(i, k=l,

^ i k “ y i Zk ‘ y k Z i

4)

!From "tlie expansions (3.4)» (3.10), the local coordinates
aj

of the generator TZ near the generator y z of the

congruence are found to be represented by the series
u} \2 = 1 + (bf+ ^ )au + (a +J'Jav
+ (o(v -f-

+

+ 2bR+n' )au2

+ a b T + aoC + JL<T +/5 Z + b f<P)4UAv

+ £(<fv + m + 2 a<T-f-<f2 ) av2 + . t.,
(12.2}<

o ^ iv1 A u 2 - Jc Y a v2 + ...,
13
- c* ^ u + J(q,+ 2c ,oC ) au 2 + c* (a-f )auav + •. •,
14
^ 2 3 " “ cav - c(b’-fcjC )auav - J(p-t2c<T) av2 + • • M
601^2= " io ?/3 A u2 H- Jq AV 2

We observe that the complex

^ 1 3 — 0 is special and

consists of all lines intersecting the line z £ . Similarly,
the complex

^ 4 2 “ ^ consists of all lines intersecting

the line yrj> ; and the complex

^ 3^ = 0 consists of all

lines intersecting the generator yz.

The lines common to

all three of these complexes form two flat pencils, one
with its center at the point y and lying in the tangent
plane of the surface S 2 at the point z, and the other with
its center at z and lying in the tangent plane of Sy at y.
The lines of these two pencils are called the central ray_s_
of the generator yz.

- 54 -

The central rays can also he characterized in another
way.

If we write the equation

(12.3)

a34<w12 + a ^ «>13 + a23 Ct,1 4 + a14^23 + a13 “V
+ a12 ^ 3 4 = °

of a general in ear complex and demand that it be satisfied
by the series (12.2) for

identically in

a u

,

at as

far as the terms of the first degree, we obtain the condi­
tions a^4 = a 23 =

= 0.

Thus the equation of the most

general linear complex having first order contact with
the congruence yz along a generator yz is found to be
(12.4)

a42 ^13 + ai3 °^42 + ai2 ^34 =

Such a complex contains the generator yz and a consecutive
generator of every ruled surface in the congruence and
containing yz.

There are obviously ©02, such linear

complexes; all of them have in common the central rays of
the generator yz and no other lines.
If we go on and seek to determine ^Linear complex
having second order contact with the congruence yz along a
generator yz, we find that such a complex exists if, and
only if, the congruence is a I congruence.

In this case

its equation is
(12.5)

q

oj1 3

+c r

= o,

and it is called the osculating linear complex along the

- 55 -

generator yz of the W congruence yz.
There is a unique complex associated with one focal
point y of a generator yz of a congruence yz in the follow­
ing manner.

The complex contains yz and all its central

rays through y.

Moreover, for every ruled surface of the

congruence through yz, the complex contains also the
generator YZ consecutive to yz and all its central rays
through its focal point Y consecutive to y.

This complex

is called the associated linear complex of the focal point
y of the generator yz of the congruence.

There is, of

course, a complex similarly associated with the other
focal point z.

In order to find the equation of the

associated linear complex of the point y we proceed as
follows.

A linear complex containing the line yz (1, 0, 0,

0, 0, 0) must have a ^ = 0.

A central ray through the

point y(l, 0, 0, 0) and an arbitrary point (0, h, 0, k) on
the line z 5 has coordinates (h, 0, k, 0, 0, 0), and belongs
to the complex in case also a23=0.
show that the complex

The series (12.2)

contains an arbitrary generator YZ

consecutive to yz in case also
complex now has the form (12.4)*

equation of the
We proceed to calculate

the coordinates of an arbitrary central ray of a neighbor­
ing generator YZ through the point Y.

The local coordinates

of the points Y, Z are respectively given by the series
(3.4), (3.10).

The local coordinates of the point Zu on

- 56 -

the u-tangent of the surface Sz at the point z are found
similarly, by expanding

2^ in powers of

a u

, av,

to be

given by certain series which turn out to be precisely
the partial derivatives of the series (3.10) with respect
to

a u

:

x 1 - m* a u +( r u + dL r ) a v + ...,

(12.6 )

J

x2 = b f+n* a u + ( / u +
x^ =

pY +

b* <T ) a v t ...,

p f a u + •..,

x, = c 1 + q 1 A U -f-C*J"AV + ...
Hr

*

The local coordinates of a point hZ

can easily be

written, and then the needed line coordinates of the
central ray joining the point Y to the point hZ + k ^ are
found to be
(*, kp* a u + ..., *, *, -kc^AU + ..., kcc' a v + ...).
These coordinates satisfy equation (12.4) identically in
h, k and to terns of the first degree in
a1 2 -°,

a42®' " a13°,|S

a u

, av

in case

°*

Thus the local equation of the associated complex of the
point y of the generator yz is found to be
(12.7)

e f/3

-t p*

0.

Similarly, the equation of the associated complex of the
point z is

These complexes are the same and are the osculating linear
complex if, and only if, the congruence is a W congruence*
If the congruence is not a W congruence the associated
complexes (12*7)» (1 2 .8 ) are distinct and determine a
pencil of linear complexes, whose equation can he written
in the form
M c ’/S 60^.3 "^P* ^ 4 2 ^+ k ((la^i3 + c
The special complexes of this pencil are given by
(he 'ft -t kq) (hpf -t kc Y ) = 0,
and consequently their equations are
respectively.

^ ^ 2 - 0 ancl c°\^- 0

The first of these, as we have already

observed, consists of all lines intersecting the line y 7 ,
and the second bears the same relation to the line z ^ •
Therefore the two lines y 7 , z ^ are the directrices of
the congruence of intersection of the associated complexes
(12.7), (12*8).

These two lines are sometimes called the

principal focal rays of the generator yz, all of the focal
rays being the tangents of the surface Sy at the point y
and the tangents of Sz at the point z*

- 5S -

BIBLIOGRAPHY

1. P. 0. Bell, A study of curved surfaces by means of
certain associated ruled surfaces. Transactions
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(1939), pp. 389-409*
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coppia dj elementi superficial! del 2? ordine.
Bollettino della TJnione Matematica Italiana, vol.
14(1935), pp. 237-2433. P. Buzano, Invariante proiettivo di una parti col are
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Unione Matematica Italiana, vol. 14(1935), pp.
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5. 0. C. Hsiung, Pro.iective differential geometry of a
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6. _______ , A study on the theory of conjugate nets.
American Journal of Mathematics, vol. 69(1947),
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7. E. P. Lane, Projective Differential Geometry of Curves
and Surfaces, The University of Chicago Press,
1932.

- 59 -

8. E. P. Lane and M. L. MacQueen, The curves of a conjugate
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9- G. Tzitzeica, Sur certains r6seaux conjuges. Gomptes
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Waelsch, Zur Infinitesimalgeometrie der Strahlencongruenzen und Fiachen. Sitzungsberichte der
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13. _______ , Geometrical significance of isothermal conjugacy.
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