be the corresponding typical representation of the assigned
boundary values of S ?R is a typical representation of
r=l
the boundary values of IT.
The moduli of periodicity of U may arise through two sources : (1)
arbitrarily assigned real moduli of periodicity at the 2p cross-cuts of the
canonical system (§ 181), that are necessary to resolve the original surface into
one that is simply connected: (2) the various moduli 9i {2'TriBr), arising from
the infinities Cr in the surface, the occurrence of which infinities renders these
additional moduli necessary for the various additional cross-cuts that must be
made in resolving the surface. Then U has all these moduli as its moduli of
periodicity, it is finite everywhere on the surface and, except for its moduli of
periodicity, it is uniform and continuous on the surface; hence it is a function
uniquely determinate, which is a constant if all the moduli be zero.
It therefore follows that the determination of u is unique, that is, that a
real function u on the Riemanns surface is determined by the general conditions
at all points on the surface except infinities, hy the assignment of specified forms
of infinities at isolated points, and hy the possession of arbitrarily assigned
moduli of periodicity at the cross-cuts which would have to be made in order
to resolve the surface into one that is simply connected. And, when all the
moduli are zero, the real function u is uniform.
Now w, = u-\- iv, is determined by u save as to an arbitrary additive
constant. Hence, summarising the preceding- results, we infer the existence
of the following classes of functions on the surface : —
(A) Functions which are finite everywhere on the surface and, except
at the lines of the cross-cuts which suffice to resolve the surface
* The form of
) = - BrAr'.
J D
And the value of the integral for the combination of the two edges of any
cross-cut c is zero.
Hence summing for the whole boundary of the resolved surface, we have
[pdQ = I {ArB; - BrAr'),
J r=l
and therefore ' ■
subject to the assigned conditions.
This theorem is of considerable importance : and the conditions, subject
to which it is valid, permit P and Q (or either of them) to be real or complex
potential functions of oc and y or to be a function of z.
231. As a first application, let P and Q be real potential functions such
that P -I- iQ is a function of z, say w, evidently a function of the first kind.
Let its moduli for the cross-cuts be
cog + ivg at tts, for s = 1, 2, ... , _p ;
and Wg+ivs at &«, for s = l, 2, ...,p.
Since P -F iQ is a function of ^ -1- iy, we have, by §§ 7, 8,
dP^dQ _dP^dQ
dx dy ' dy dx'
231.] SYSTEM OF MODULI 501
The double integral then becomes
which cannot be negative, because P is real ; it is a quantity that is positive
except only when P (and therefore w) is a constant everywhere. In the
present case
Ar = (&>2,i, 6^2,2, •••> «2,2f))) •••> ('"g,!. ft>q,-2, •■■,c^q,2p)-
The modulus of wv at the cross-cut G^n has its real part denoted by &)^_,„ :
when the modulus is divided into real and imaginary parts, let it be
^r, m 1 ^^ r, m •
If any set of g arbitrary complex constants be denoted by Ci, ..., Cq, where
Cg is of the form Og + i^g, then, at the cross-cut Cm, the real part of the
modulus of 2 Cr^i'r is the real part of 1 Cr((Or,m + *'&>';•, wX that is, it is equal to
r=l r=l
Ol<«l,»i+ ••• + ^q(^q,m- 0lf^\,m— ••• — 0q(^'q,m,
holding for ??i = l, 2, ..., 2p, and therefore giving 2^) expressions in all.
Now let a set of real arbitrary quantities A^, A.^, ..., A^p be assigned as
the real parts of the moduli of periodicity of a function of the first kind,
which is uniquely determined by them ; and consider the equations
-^1= «!&>!, 1-1- a2&)2,l+ ... -I- OfgW^^l — ZSj ft)'i_i — yS2 <»'2,1— ••• —y^^tu'j^i 1
A.p= ai2,22)+ ••• + '^qfOq,2p - ^lO)\,op — ^2(^'2,2p - ••• — ^q<»'q,
2p)
First, let qn, C0i2, , 0)ip
pi > f"p2 > ) f"pp
cannot vanish : for otherwise by taking constants Cj, ...,Cp proportional to the
p
first minors, we should obtain a function S CgiVg, having all its moduli for
the cross-cuts a-^, ..., ap zero and therefore, by § 231, merely a constant, so
that Wi, ..., iVp would not be linearly independent. Hence A does not vanish.
Next, we can choose constants c so that the moduli of periodicity vanish
p
for the function S CgWs at all the cross-cuts a, except at one, say a,., and
that there it has any assigned value, say iri. For, solving the equations,
= Cia>s,i + c.20)s^o + ... + CpCOg^p, (for s = 1, 2, .,., p, except r);
Tri = Ci (Or, 1 -f Co &)r, 2 + • • • + Cp(Or,p,
the determinant of the right-hand side does not vanish, and the constants c,
say Cr^i, c,.,2, ..., Cr,p, are determinate. The function Cr^iW-^ + 0^,2^2 + •■• + Gr,pWp,
say Wy, then has its moduli zero for ttj, ..., ttr-i, ^r+u ..., a^: it has the
modulus iri for a^; it has moduli, say 5^,1, ..., B^^p at b^, ...,bp respectively.
And the function is determinate save as to an additive constant.
This combination can be effected for each of the values 1, ...,p of r:
and thus p new functions will be obtained. These p functions are linearly
independent : for, if there were a relation of the form
GjW^ + C2W2 + . . . + GpWp = constant,
p
the modulus of the function 2 GrWr at the cross-cut a^ would be zero
r=l
because the function is a constant ; and it is Cs'jri, so that all the coefficients
would be zero.
508 NOEMAL FUNCTIONS OF THE FIRST KIND
The functions W, thus obtained, have the moduli : —
[235.
«l
«2
dp
&1
&2
&P
w.
7^^■
A.1
A, 2
a:p
^2
7^^
I
A,l
^2,2
-^2,p
w.
TTl
-^p.l
^P,2 !
^P,P
These functions are called normal functions of the first kind : they are a
complete system linearly independent of one another, and are such that
every function of the first kind is, except as to an additive constant, a linear
combination of constant multiples of them.
The quantities B are not completely independent of one another. Since
Wj, Wf are functions of the first kind we have, by § 232,
>• = !
which, for the normal functions, takes the form
•jriBjf — iriBj'j = 0,
that is, Bjf = Bj'j. Hence the moduli B with the same pair of integers for
sufiix are equal to one another.
This is a first relation among the moduli. Another is given by the
following theorem : —
Let B„i^n = Pm,n + 'i(^m,n, (sO that Pm,n = Pn,my Cl^d CT^^n = (^7i,m) '• then, if
Ci, ..., Cp he any real quantities, the expression
pnCi^ + 2/!3i2CiC2 + p^Co'+ ... + PppCp%
is negative, unless the quantities c vanish together.
The function c-iW-^ + c..W and the normal integral is
TriFjz)
ttK'
The other period of this function is — ^rfr, which, when k is real and less than
unity, is a negative quantity ; it is the value of p^ and satisfies the condition
that puCi is negative for all real quantities c.
236. It has been proved that functions exist on a Riemann's surface,,
having assigned algebraic infinities and assigned real parts of its moduli
of periodicity, but otherwise uniform, finite and continuous. The simplest
instance of these functions of the second kind occurs when the infinity is an
accidental singularity of the first order.
Let the single infinity on the surface be represented by ^^ = c : let Ec {z}
be the function having 5 = c as its algebraical infinity, and having the real
parts of its moduli of periodicity assigned. If Eg {z) be any other function
with that single infinity and the real parts of its moduli the same, then
Eci^) — Ec (z) is a function all the real parts of whose moduli are zero; it
does not have c for an infinity and therefore it is everywhere finite : by § 231, it
is a constant. Hence an elemeyitary function of the second kind is determined,
save as to an additive constant, by its infinity and the real parts of its moduli.
Again, it can be proved, as for the special case in § 208, that an elementary
function of the second kind is determined, save as to an additive function of
the first kind, by its infinity alone : hence, if E (z) be any elementary function,,
having its infinity at ^^ = c, we have
E (z) = Ec(z) + \W,+ ... + \pWp + A,
where \, ...,\p,A are constants, the values of which depend on the special
function chosen. Let E^z) have TriC^, ..., TriCp for its moduli at the cross-
cuts «!,..., a^ respectively : and let the function E (z) be chosen so as to-
510 NOEMAL FUNCTION OF THE SECOND KIND [236.
have all its moduli at a-^, ...,ap equal to zero : then \,. = — 6',. and £" (z) is
given by
The special function of the second kind, which has all its moduli at the cross-
cuts tti, ..., Op equal to zero, is called the normal function of the second kind.
It is customary to take unity as the coefficient of the infinite term, that is,
the residue of the normal function.
This normal function is determined, save as to an additive constant, by its
infinity alone. For HE {z) and E'{z) be two such normal functions, the function
E{z) - E'{z)
is finite everywhere; its moduli are zero at ttj, ..., o^; hence (§231) it is a
constant.
Normal functions of the second kind will be used later (§ 240) in the
construction of functions with any number of simple infinities on the surface.
Let the moduli of this normal function E (z) of the second kind be
Bj, ..., Bp for the cross-cuts h^, ..., bp. Then applying the proposition of
§233 and considering the integral [EdW,., we have A^= ... =Ap = ; also
Ai = ... = A r—i — A r^i = ... = Ap = 0,
and Ar = 7ri. The relation therefore is
„ . ^ .fdWr\
- Br-m -h 27^^ I -~ = 0,
V djZ J z=c
where, in the immediate vicinity of ^^ = c,
E{z) = j--^+p{z-c),
p being a converging series of positive powers. Thus
dW
or, as —j-^ is an algebraic function (§ 241) on the surface, the periods of a
normal function of the second kind at the cross-cuts h ai^e algebraic functions
of its single infinity.
In th« case of the Jacobian ellii^tic integrals, the integral of the second kind has at
z= 00 an infinity of the first order in each sheet (Ex. 8, § 199). The moduli of this integral,
denoted by E (z), are 4E and 2i (K' - E') for a^ and b^ respectively ; hence the normal
integral of the second kind is
F{z) being the (one) integral of the first kind. This is the function Z{z). Its modulus is
zero for aj ; for hi , it is
^i{K'-E')-^^2iK',
which is -r^{KK' - E' K - EK'), that is, the modulus is —^-
237.] NORMAL FUNCTION OF THE THIRD KIND 511
237. The other simple class of function, which exists on a Riemann's
surface with assigned infinities and assigned real parts of its moduli, is that
which is represented by the elementary integral of the third kind. It has
two points of logarithmic infinity on the surface*, say Pi and P^; let these
be represented by the values Ci and Ca of z. On division by a proper constant,
the function, which may be denoted by Ilia, takes the forms
- log {z - Ci) + pi{z- Ci), + log {z - Ca) + i?2 (^ - C2),
in the immediate vicinities of Pi and of Pg respectively, where p^ and p2 are
converging series of positive integral powers.
The points Pi and P2 can be taken as boundaries of the surface, as in
Ex. 7 in § 199. A cross-cut fi-om P2 to Pi is then necessary for the resolution
of the surface : and the period for the cross-cut is ^iri, being the increase of the
function in passing from the negative to the positive edge of the cross-cut.
Then with this assignment of infinities and with the real parts of the
moduli at the cross-cuts tti, ..., a^, 61, ... , 6^ arbitrarily assigned, functions Ilia
exist on the Riemann's surface.
As in the case of the function of the second kind, it is easy to prove that a
function Oio of the third kind is determined, save as to an additive constant, by
its two infinities and the assignment of its moduli : and that it is determined,
save as to an additive function of the first kind, by its infinities alone.
Among the infinitude of elementary functions of the third kind, having
the same logarithmic infinities, a normal form can be chosen in the same
manner as for the functions of the second kind. Let IIio be an elementary
function of the third kind, having Pi and Po for its logarithmic infinities :
let its moduli of periodicity be ^iri for the cross-cut P1P2; Trt'Ci, ..., iriCp for
ai, ..., ap respectively; and other quantities for h^, ..., hp respectively. Then
^,,= Ii,,-G,W,-...-GpWp
is an elementary function of the third kind, having zero as its modulus of
periodicity at each of the cross-cuts a^, ..., ap. This function is the normal
form of the elementary function of the third kind.
If -5712' and ffiTia be two normal elementary functions of the third kind with
the same logarithmic infinities and the same period liri at the cross-cut
PiPo, then
■SJia — "STia
is a function without infinities on the surface ; its modulus for Pi Pg is zero,
and its modulus for each of the cross-cuts ai, ...,ap is zero; and therefore
* The representation of a single point on the Riemann's surface by means solely of the value
of z at the point will henceforward be adopted, without further explanation, in instances when it
cannot give rise to ambiguity. Otherwise, the representation with full detail of statement will
be adopted.
512 MODULI OF NOKMAL ELEMENTAEY FUNCTION [237.
it is a constant. Hence a normal elementary function of the third kind is
determined, save as to an additive constant, by its infinities alone.
Ex. The sum of three normal elementary functions of the third kind, having as
their logarithmic infinities the respective pairs that can be selected from three points,
is a constant.
238. A relation among the moduli of an elementary function of the third
kind can be constructed in the same way as, in § 233, for the function of the
second kind.
Let the surface be resolved by the 2p cross-cuts a^, ..., a^, \, . . . , hp and by
the cross-cut P^P^, joiiiing the excluded infinities of an elementary function
Ilia of the third kind. Let iv be any function of the first kind ; then over the
resolved surface, we have
dx dy dy dx
everywhere zero; and therefore jlii^dw round the whole boundary of the
resolved surface is zero, as in § 233.
Let the moduli of Ilja be A^, ..., Ap, B^^, ..., Bp, and those of w be
J./, ..., Ap, B^, ..., Bp for the 2^ cross-cuts a and h respectively.
The whole boundary is made up of the two edges of the cross-cuts a, the
two edges of the cross-cuts h, the two edges of the cross-cut P^P^, and the
small curves round Pj and Pg.
The sum of the parts contributed to jTl^zdw by the edges of all the cross-
cuts a and h is, as in preceding instances,
i{A,B:-A:B,).
The direction of integration along PjPa that is positive relative to the area
in the resolved surface is indicated by the arrows ; the
portion of ^W-^^dw along the two edges of the cut is J , ^ ^
^2 r ^1 .
ITia+cZw; 4- Yl^^rdw
Ci .' C2 Fig. 83.
ri, ..., Bp
as its moduli for the cross-cuts a-^, ..., Up, b^, ..., 6^; and let 1134 be another
elementary function with logarithmic discontinuities at Cg and C4, with ^iri as
its modulus for the cross-cut C3C4, and with A-^, ,.., Ap, 5/, ...,Bp' as its
moduli for the cross-cuts a-^, ..., ap, b^, ...,bp.
Then when the infinities are excluded and the /^nK^ i-W>
surface is resolved so that both IT 12 and 1134 ^ G O + m
are uniform finite and continuous throughout f Ar r
the whole surface, we have '^^ + Hi d"^
9 II 12 51134 _ 9II34 dU-^2 ^ Q Fig. 84.
dx dy dx dy '
everywhere in the resolved surface ; and therefore, as in the preceding
instances, / 012^^1134 round the whole boundary vanishes.
The whole boundary is made up of the double edges of the cross-cuts a
and the cross-cuts 6, and of the configuration of cross-CTits and small curves
round the points. The modulus of both Ilia and 1134 for the cut AG is,
zero ; the modulus of II12 for the cut C3C4 is zero, and that of 1134 for the cut
C1C2 is zero.
The part contributed to /IIi2C^n34 by the aggregate of the edges of the
p
cross-cuts a and 6 is S {AgBg — AgB^, as in preceding cases.
s=\
F. F. 33
514 INTERCHANGE OF ARGUMENT [238.
The part contributed by the small curve round Ci is zero, because the
limit, for z = d, of (z — d) ITis -j^ is zero. Similarly the part contributed by
the small curve round Ca is zero.
The part contributed by the two edges of the cross-cut C1C2 is
= 27^^• r dU^ = 27ri [U^ (c,) - U,, (c,)}.
J c,
The part contributed by the two edges of the cross-cut AG is
the subject of integration does not change in crossing from one edge to the
other, and therefore this part is zero.
For points on the small curve round C3, we have
dJls4 = — — \- p(z — C3) dz,
Z C3
where p is a converging series of integral powers of z— Cs: and therefore for
points on that curve
U^^dUsi = '-^-^ dz + q(z- Cg) dz,
where q (z — Cs) is a converging series of positive integral powers of z — Cg.
Hence the part contributed to JYIi^dns^ by the small curve round C3 in the
direction of the arrow, which is the negative direction for integration relative
to C3, is 27rin 12(03).
Again, for points on the small curve round C4, we have
dz
dUsi = h Pi{z — Ci) dz.
Proceeding as for C3, we find the part contributed to {^y-^d'U.-.i^ by the small
curve round C4, which is negatively described, to be — 27rini., (C4).
Lastly, the sum of the parts contributed by the two edges of the cross-cut
C3C4 IS
'*ni,(^n34++ [''ni2c^n34-
238.] AND PARAMETER 515
But though 1134 has a modulus for the cross-cut C3C4, its derivative has not a
modulus for that cross-cut : we have dU..^-^/dz = dlis^jdz, and therefore the
last part contributed to jYl^^d^M vanishes.
The integral along the whole boundary vanishes ; and therefore
I (^,B; - A^B,) + 27ri {U^ (cs) - n,, (c,)] + 27rm,, (c,) - 27riU,, (C4) = 0,
a relation between the moduli of two elementary functions of the third kind.
The most important case occurs when both of the functions are normal
elementary functions. We have A^, ..., Ap zero for tn-ja, and Ai, ..., Ap zero
for 13-34 ; and the relation then is
tD-34 (Ca) - -3734 (Ci) = tSTi.^ (C4) - tn-12 (Cs),
which is often expressed in the form
re-2 rc-i
- Ci •' C3
the paths of integration in the unresolved surface being the directions of
cross-cuts necessary to complete the resolution for the respective cases.
Hence the normal elementary integral of the third kind is unaltered in value
hy the interchange of its limits and its logarithmic infinities.
Ex. 1. Denoting by Ei and E^ the normal elementary integrals of the second kind
which have their single simple infinities at Cj and c^ respectively, shew that the value ,
of —j^ at Co is equal to the value of —r^ at c, .
dz ' ^ dz ^ -
Ex. 2. Denoting by E^ the normal elementary integral which has its single infinity
,at C3, prove that the value of '^ at c^ is E-i{c2)- E^{ci).
239. From the simple examples, discussed in § 199 and elsewhere, it has
appeared that when a function w is defined as the integral of some function
of z, the integral being uniform except in regard to moduli of periodicity, a
process of inversion is sometimes possible whereby z becomes a function of w,
■ either uniform or multiform. But in all the cases, in which z thus proves to
be a uniform function, the number of periods possessed by w is not gTeater
than two; and it follows, from § 110, that, when w possesses more than two
periods, z can no longer be regarded as a uniform function of w.
A question therefore arises as to the form, if any, of functional inversion,
when w has more than two independent periods and when there are more
functions w than one.
Taking the most general case of a Riemann's surface of class p, let
w-^, w^, ..., Wp denote the jp functions of the first kind. Let there be q inde-
pendent variables z-^^, ..., Zq, where q is not, of initial necessity, equal to p;
33—2
516 PROBLEM OF [239.
and, by means of any q of the functions of the first kind, say Wi, ...,Wq, form
q new functions, also of the first kind, and defined by the equations
-y,. = lU^ {z-^ + Wr (^2) -\- ... -\-Wr (Zq),
where r = 1, 2, ..., q. We make the evident limitation that q is not greater
than p, which is justifiable from the point of view of functional inversion.
Then the functions v,. are multiform on the surface with constant moduli of
periodicity ; they have the same periods as Wr, say t»,-,i, cor,2, •••> c^r,2p-
The various values of Wr (^m) differ by multiples of the periods : so that, if
Wr(zm) be the value for an exactly specified ^^-path (as in § 110), the value
for any other ^j^-path is
This being true for each of the integers w = 1, 2, ... , q, it follows that, if
Q
ms= % nm,s, {s = l,2,...,2p),
m = l
q
and if v.y be the value of Z w,. (z^n) for the exactly specified paths for z^, •••, 2q^
then the general value of v,. for any other set of paths for the variables is
Vr + mift),.^i 4- 1112,(0 r, 2. + ... + ^2pf^r,2p>
holding for r = 1, 2, ..., q. The integers 7im,s, and therefore the integers Wg,.
are evidently the same for all the functions v.
The reason which, in the earlier case (§ 110), prevented the function w from
being determinate as a function of z alone was, that integers could be determ-
ined so as to make the additive part of w, dependent upon the periods, less
than any assigned quantity however small : say less than an infinitesimal
quantity. It is necessary to secure that this possibility be excluded.
Let (0\^fi = a^^fj, + i^\^fj,, where the quantities a and /3 are real: then we
have to prevent the possibility of the additive portions for all the functions v
being infinitesimal. In order to reduce the additive part to an infinitesimal
value for each of the functions v, it would be necessary to determine integers
oiii, niz, ..., m^p so that the 2q quantities
rriyfir,! + vn^^r,2 + • • ■ + 'ni2p^r,2p
for r = 1, .,., q, all become infinitesimal.
If q be less than p, the 2p integers can be so determined. In that case^
the general possibility of functional inversion between the q functions v and
the q variables z would require that the quantities z are so dependent upon
the quantities v that infinitesimal changes in the latter, carried out in an
239.] INVERSION 517
infinite variety of ways and capable of indefinite repetition, would leave the
quantities z unchanged. The position, save that we have q variables instead
of only one, is similar to that in § 110 : we do not regard the functions v as
having determinate values for assigned values of z-^, ..., Zg, but the values of
Vi, ...,Vq are determinate, only when the paths by which the independent
variables acquire their values are specified. And, as before, the inversion is
not possible.
If q be not less than p, so that it must in the present circumstances be
equal to p, then the 2p integers cannot be determined so that the 2p quanti-
ties all become infinitesimal. They can be determined so as to make any
2p—l of the quantities become infinitesimal ; but the remaining quantity
is finite as, indeed, should be expected, because the determinant of the
constants a. and /3 is different from zero*.
If then there be p variables Zi, ..., Zp, and p functions Vi, ..., Vp, defined
by the equations
Vr = lUr (^i) + W,. (Z.2) + ...+ W,. (Zp),
for ?^ = 1, 2, ..., j^, then the values of the functions v for assigned values of
the variables z, whatever be the paths by which the variables attain these
values, are of the form
for r = 1, 2, ..., p ; and it has been proved that the 2p integers m cannot be
determined so that all the additive parts, dependent upon the periods, become
infinitesimal. Hence the functions v-^, ..., Vp are, except as to additive
multiples of the periods (the numerical coefiicients in these multiples being
the same for all the functions), uniform functions of the variables z^, ..., Zp-,
and they are finite for all values of the variables. Conversely, we may regard
the quantities z as functions of the quantities W], ..., Vp, which have 2p sets
of simultaneous periods Wi^g, &>2,6., ..., &)^,s for s = l, 2, ..., 2p: that is, the
variables z are 2j9-ply periodic functions oi p variables v^, ..., Vp. This result
is commonly called the inversion-pi^ohlem for the Abelian transcendents.
In effecting the inversion of the equations
dvi = u\ {z-^) dzi + ■z^i' (^s) dz.2+ ... + Wi (zp) dzp\
dvp = lUp {z-) dzi + tUp {z.^ dz^-Jr ... + w/ (Zp) dzp]
the actual form, which is adopted, expresses all symmetric functions of
the quantities ^■j, ...,% as uniform functions of the variables, so that, if
z-^, Z.2, . . . , Zp he the roots of the equation
(/) {Z) = ZP + P,Zp-' + P,ZP-' + ...+Pp = 0,
* The 2p potential-functions, arising from the p functions w, would otherwise not be linearly
independent.
518 ABELIAN FUNCTIONS [239.
then* Pi, ..., Pp are uniform multiply-periodic functions of the variables
Vi, ...,Vp. Consequently, all rational symmetric functions of z-^, ...,Zp are
uniform multiply-periodic functions oi v-^, . . . , Vp.
Frequent reference has been made to the functions determined by the equation
w^- R(z) = w^ — {z-ao) (z- cci) ...{z- a2p) = 0.
f U (z)
It has been proved that an integral of the form I — ^ dz is an integral of the first kind,
provided U {z) be any polynomial function of degree not higher than ^-1, and that
the otherwise arbitrary character of U {z) makes it possible to secure the necessary
p integrals by allowing the suitable choice of the coefficients. "Weierstrass takes the
equations, which lead to the inversion, in the following formt : —
The constants a are different from one another and can have any values : and it is
convenient to take
P {x) = {x- ai) {x -a3)...{x- a^p-i),
Q{x) = {x-aQ){x-a2)...{x-a.2p-2){x-ao^p\
so that P{x) Q{x) — R{x). If the coefficients a be real, it is assumed that
ao > «i > 0^2 > ••• > «^2p-
The equations which give the new variables are
^ P (0i) dzr P{z.,)dz2 , ^ P(zp)dzp
au-,= , -I- , -r -t- -
(^1 -ai)'jR {zx) (S2 -ai)\/R (z^) (zp - ay) s/R (Zp)
du = ^ ^^'^ ^^1 + Pi^2)dzz _^ _^ P (zp) dzp
(^1 - as) Vii (si) (02 - «3) "^R (^2) {z,p - as) "-' R (^p)
P{z^)dz^ , Piz2)dz2 P{zp)dzp
a\ip = ■ H ; — ,„ , . + H
(^1 - «2p-i) "^R (%) (22 - ^2^-1) "^R (22) {h - «2p-i) ■^R (s),
and when integration takes place, the arbitrary constants are defined by the equations
Wi, ?<2; •••5 Wp = (with periods for moduli),
when Sj, 235 •••) %=«!) «3) •••> «2p-i respectively.
The p variables z are the roots of an algebraical equation of degree jo, the coefficients in
which are (multiply-periodic) uniform functions of the variables u. The functions, arising
out of the equations in this form, are discussed J in Weierstrass's two memoirs, just
quoted.
Note 1. The results thus far established in this chapter lie at the basis of the theory
of Abelian functions. The fuller establishment of that theory and its development
are beyond the range of the present treatise.
So far as concerns the general theory, recourse must be had to the fundamental
memoirs of Abel, Jacobi, Hermite, Riemann and Klein, and to treatises, in addition to
* For further considerations see Clebsch und Gordan, Theorie der AheVschen Functionen,
Section vi.
t Equivalent to that given in Crelle, t. lii, (1856), pp. 285 et seq. ; it is slightly diiierent from
the form adopted by him in Crelle, t. xlvii, (1854), p. 289.
X Some of the results are obtained, somewhat differently, in a memoir by the author, Phil.
Trans., (1883), pp. 323—368.
239.] UNIFORM FUNCTIONS ON RIEMANN'S SURFACE 5l9
those by Neumann and by Clebsch and Gordan already cited, by Prym, Krazer, Konigs-
berger, Briot, and Stahl. The most comprehensive of all is Baker's treatise Abel's theorem
and the allied theory^ including the theory of the theta functions, (Cambridge, 1897).
Moreover, as our propositions have for the most part dealt with functions of only a
single variable, it is important in connection with the Abelian functions to take account
of Weierstrass's memoir* on functions of several vaciables.
Note 2. We have discussed only very limited forms of integrals on the Riemann's
surface : and any professedly complete discussion would include the theorem that \iv'dz,
where ^o' is a general function of position on the surface, can be expressed as the sum of
some or all of the following parts : —
(i) algebraical and logarithmic functions ;
(ii) Abelian transcendents of the three kinds ;
(iii) derivatives of these transcendents with regard to parameters ;
but such a discussion is omitted as appertaining to the investigations relative to Abelian
transcendents. Supplementary Notes will be found at the end of this Chapter XVIII.,
giving an account of Abel's theorem, and indicating a mere beginning of the theory of
Abelian transcendents.
For the particular case in which the integral iio'dz is an algebraical function of 2, see
Briot et Bouquet, TheoHe des fonctions elliptiques, (2™« e'd.), pp. 218- — 221 ; Stickelberger,
Crelle, t. Ixxxii, (1877), pp. 45, 4fi ; and Humbert, Acta Math., t. x, (1887), pp. 281—298,
by whom further references are given.
240. There are functions belonging to class {B) in § 229, other than
those already considered. In particular, there are functions with assigned
infinities on the surface and with the real parts of all their moduli of
periodicity for the canonical system of cross-cuts equal to zero. But it
does not therefore follow that all the moduli of periodicity vanish ; in order
that their imaginary parts may vanish, so as to make the moduli of
periodicity zero, certain conditions would require to be satisfied.
We shall limit the ensuing discussion to some sets of these functions
with zero moduli, and shall assign the conditions necessary to secure that
the moduli shall be zero. We shall assume that all their infinities are
algebraic ; the functions are then uniform everywhere on the surface, and,
except at a limited number of isolated points, where they have only
algebraic infinities, are finite and continuous. They are, in fact, algebraic
functions of z.
Two classes of these functions are evidently simpler than any others.
The first class consists of those which have a limited number, say m, of
isolated accidental singularities and which are not infinite at any of the
branch-points ; the other class consists of those which have no infinities
except at the branch-points. These tAvo classes will be briefly discussed
in succession.
* First published in 1886 ; Ahliandhingen aus der Functionenlehre, pp. 105 — 164 ; Ges. Werke,
t. ii, pp. 135 — 188. See also the author's Lectures introductory to the theory of functions of tivo
complex variables (Camb. Univ. Press, 1914).
520 UNIFORM FUNCTIONS IN TERMS OF [240.
Let w be a uniform function having accidental singularities at the
points Ci, ...,Cm and no other infinities; and for simplicity, assume* that
each of them is of the first order. Also let the normal function of the
second kind, having c,. for its sole infinity, be Z^. Then
where /Si, ..., ^m are constants at our disposal, is a function, having infinities
of the same class and at the same points as tv has; the function is otherwise
finite everywhere on the surface and therefore, by properly choosing the
constants /S, we have the function
VJ-{^rZ,+ ...+^,r,Z„,)
finite everywhere on the surface, so that it is a function of the first kind.
Now because its modulus vanishes at each of the cross-cuts a in the
resolved surface, it is a constant, so that
tU=^,Z,+ ...+|3n^Z,,,+0o■
The modulus of w is to vanish at each of the cross-cuts 6^. Let (f)r(z)= , ^ ,
so that r {c.^ + ...+ ^mA (Cm) = 0,
an equation which must hold for all the values r = 1, ...,p.
When the quantities c represent quite arbitrary points, thei^e must be
at least p+ 1 of them ; otherwise, as the equations are independent of one
another, they can be satisfied only by zero values of the constants /5, a result
which renders the uniform function evanescent. If m > p, the equations
determine p of the coefficients /S linearly in terms of the remaining on - p :
when these values are substituted, the resulting expression for w contains
^Yi — p -^ I constants, viz., the remaining m — p constants yS, and the constant
/3o. The coefficient of each of the m — p constants /3 is a function of z, which
has p+l accidental singularities of the first order, p of which are common
to all the functions, so that w then is an arbitrary linear combination
of constant multiples of m — p functions, each of which possesses ^3 + 1
accidental singularities and can be expressed in the form
A,. (2;)= Zi, Zo, , Zp, Zp^r
<^l (Ci), ! (C2), , ! (Cp), 4>i (Cp+r)
4>2{Ci), 2{Cp)- (p-2(Cp+r)
^p{Ci), <\>p{CoX (i>p{Cp), (f)piCp+r)
If a pole, say at cj, is of order s, then /Si^i would be replaced by
OCi
in the expression for tv ; and so for other cases.
71+72 g^+...+-,.g,^.
240.] NORMAL FUNCTIONS OF THE SECOND KIND 521
When the quantities c are not completely arbitrary, but are such that
relations among them can be satisfied so as no longer to permit the preceding
forms to be definite, we proceed as follows.
The most general way in which the preceding forms cease to be definite
is by the dependence of some of the equations
^,(f>r (Ci) + /3,(br (Ca) + . . . + /3,nr (Cm) =
on the remainder. Let q of them, say those given by r = l, ..., q, be
dependent on the remaining p — q, so that 0,. (Cn) = ^1, r4>q+l (^n) + ^2,rp (C„),
for 7' =1, 2, ..., q and n = 1, 2, ..., m.
The functions of the first kind W, through which the functions <^ are
derived, are a complete set of normal functions : when any number of them
is replaced by the same number of independent linear combinations of some
or all, the first derivatives are still algebraic functions. We therefore replace
the functions W-^, TTo, ..., Wq by Wi, Wg, ..., Wq, where
for r = 1, 2, ..., g, so that, for all values of z,
^r{^) = ^r{2)-A,^r(i>q+ii2)-A,^rq+2(2)- . .. - Ap^q^.^pi^)-
Hence the functions i, o, ..., ^q vanish at each of the points Cj, Ca, ..., c,„.
The original system of j) equations in ^j, ..., q+i, ..., <^p, when made
a system of equations in 3, ..., g, ^q+i, ..., (f)p is equivalent to
A % (Ci) + l3,^r (C,) + ..•+ An <^,- (C,n) = 0)
A 0s (Ci) + A 0, (C.,) + . . . + /3„, (f)s (Crn) = Oj '
for r = 1, ..., q and s = q + 1, ..., p. The first q of these are evanescent ; and
therefore their form is the same as if we had initially assumed that each of
the functions (f}^, ..., (f)q vanished for each of the points z = Ci, ..., c^, the two
assumptions being in essence equivalent to one another on account of the
property of linear combination characteristic of functions of the first kind.
Suppose, then, that q of the functions (f), derived through functions of the
first kind, vanish at each of the points Cj, ..., c^; the .number of surviving
equations of the form
/Si (fir (Ci) + /3,<^, (C.) + . . . + /3,«0r (C,n) =
is p- q, and they involve m arbitrary constants /S. Hence they determine
p — q of these constants, linearly and homogeneously, in terms of the other
7n—p + q. When account is taken of the additive constant ^o, then* the
* This is usually known as Eiemann-Roch's Theorem. It is due partly to Eiemann and
partly to Eoch ; see references in § 242.
522
RIEMANN-ROCH S
[240.
function tu contains m.—p-\-q^-l arbitrary constants ; and it is a linear com-
bination of arbitrary multiples of m—p + q functions, each having p — q--\-\
accidental singidarities of the first order, p — qof which are common to all
the functions in the combination.
The functions under consideration, being linear combinations of normal
functions Z of the second kind, have no infinities except at the accidental
singularities ; the branch-points of the surface are not infinities. And it
appears, from the theorem just proved, that there are functions having only
p — q+1 accidental singularities, each of the first order, so that the total
number is less than p + 1. A question therefore arises as to what is the
inferior limit to the number of accidental singularities that can be possessed
by a function which is uniform on the Riemann's surface and which, except
at these accidental singularities, is everywhere finite and continuous on the
surface.
Let this limit be denoted by /a ; then the p equations
A<^r(ci)+...+/3^(^,.(c^) = 0,
for r = l, 2, ...,p, must determine yu, — 1 of the constants 13 in terms of the
remaining constant /3, say, B. The function thence determined contains two
constants, viz., the surviving constant j3 and the additive constant, its form
being
A+B
i (Ci),
■Z„
z.
2 (Cm)
<^^_i(Ci),*0^_i(C2), , <^^_i(c^)
Among the points Ci, Cj, , c^, the relations
(j>i(c,), <^i(Co), , <^a(c^)
for r = 0, 1, ...,p — fX; must be satisfied, that \s,p — /u. + l relations must be
satisfied *.
Since there are fi points c among which p — /x+l relations are satisfied,
it follows that the number of surviving arbitrary constants c is, in general,
equal to fi -(p — /jl+I), that is, to 2/j, — p — 1. These occur as arbitrary con-
stants in the inferred function, independently of the two constants A and B :
so that the number of arbitrary constants, in the function with //, accidental
singularities, is 2yu, — p — 1 + 2, that is, 2yu, — p + 1.
* This result implies that the relations are independent of one another, which is the case in
general : but it is conceivable that special relations might exist among the branch-points, which
would affect all these numbers.
240.] THEOREM 523
Again, the number of infinities of a uniform function of position on a
Riemann's surface is equal to the number of its zeros (| 194), and also to
the number of points where it assumes an assigned value ; and all these
properties are possessed by any function, with which lu is connected by
any lineo-linear relation. If u be one such function, then another is
au + b
lu — 7 ,
■u — a
where a, h, d are arbitrary constants ; and therefore lu contains at least
three arbitrary constants, when it is taken in the most general form that
possesses the assigned properties.
But it has been shewn that the number of independent arbitrary con-
stants in the general form of w is 2/j, — p+l. This number has just been
proved to be at least three, and therefore
2fi-p + l^S,
or yu, ^ 1 + ^p.
Thus the integer equal to, or next greater than, \ -{- hp is the smallest
number of isolated accidental singularities that an algebraical function can
have on a Riemanns surface, on the supposition that it has no infinities at
the branch-points*.
Note. A method of decomposing rational functions on a Riemann's
surface, so that the elements are normal functions of the second kind, is
given above ; another method of constructing a rational function is as
follows.
It was seen that the number of simple poles of a rational function, when
all of them are arbitrarily assigned, cannot be less than p + 1; to consider
the simplest case, we accordingly assign ^ + 1 arbitrary points, which shall
be infinities (and the only infinities) of a rational function. Take the most
general polynomial P (w, z) of order n— 2 in w and z combined, and make it
vanish at the ^ + 1 assigned points. Also make P vanish at each of the
multiple points of the curve /= in such a way that, when the point is of
multiplicity A, for /= 0, it is of multiplicity X — 1 for P = 0; consequently
such a point counts for A,(X— 1) intersections among the points common to
/=0, P = 0. Hence the number of mtersections common to/=0, P = 0,
other than the multiple points of / = 0, and the p + l arbitrary points, is
= n{n-2)-(p + l)-^\(X-l).
* This result applies only to a completely general surface of class p. And, for special forma
of siu'face of class p, a lojver limit for /j. can be obtained; thus, in the case of a two-sheeted
surface, the limit is 2. (See Klein-Fricke, i, p. 556.)
524 CONSTEUCTION OF FUNCTIONS [240.
But, as in § 182, we have*
|9 = i(w-l)(?i-2)-2U(X-l);
and therefore the number of remaining intersections is
= n (n - 2) - ( j9 + 1) - {n - 1) {n -2) + 2p
= n + p — S.
Now a polynomial in w and z of order n-2 contains in(n-l) terms.
The number of relations among the constants, necessary to secure that
a point on a curve is of order k, is ^k(k + l); so that the number of
relations among the constants, necessary to make each. of the multiple points
of /=0 a multiple point of the proper order for P = 0, is l^\(\-l).
Since
^n{n -l)-l^\(X-l)-(n + p- 3)
= i?r (a - 1) - ^{n - l){n - 2) + p - {7i+p - 3)
= 2,
it follows that there are two independent polynomials, which can be drawn
through the multiple points of /=0, vanishing to the proper order at each
of them, and through the n+p -3 points common to /= 0, P = 0, which
are other than the p + 1 arbitrary zeros of P. Clearly P itself can be
taken as one of these polynomials; let P^ denote another independent
of P. In general. Pi does not pass through any other zero of P ; in
order to make it do so, one other relation among the constants would
be necessary, and then there would be only a single polynomial passing
through the multiple points to the proper order, through the n+p-S
points, and through the other zero of P: the single polynomial being
P itself.
Thus
_ Pi (w, z )
^' P{iv,z)'
is a function, which has the p + 1 assigned points for poles, because they
are zeros of P and not of Pj ; all the other zeros of P on the surface
are zeros of Pi to the same order,, and they therefore are not jDoles of g.
Thus a rational function has been constructed, which has ^ + 1 assigned
points as poles, and it has no other poles.
241. The other simple class of uniform functions on a Riemann's
surface consists of those which have no infinities except at the branch-
points of the surface.
They will not be considered in any detail : we shall only briefly advert
* A multiple point of order X on a curve is equivalent to i\(\ — 1) double points (Salmon's
Higher Plane Curves, § 40) : hence the aggregate of equivalent double points is SiX (\ - 1).
241.] ALGEBRAIC FUNCTIONS 525
to those which consist of the first derivatives of functions of the first kind.
This set is characterised by the theorem : —
These functions (f) (z) are infinite only at branch-points of the surface, and
the total number of infinities is 2p — 2+ 2n. For, let w (z) be the most
general integral of the first kind, and let
Near an ordinary point a on the surface we have
w (z) = w (a) + (z — a) P (z — a),
where P is a converging series that may, in general, be assumed not to
vanish for z = a] hence
(z) = P(z -a) + (z - a) P' {z- a);
that is, cf) (z) is finite at an ordinary point.
Near z= ao (supposed not to be a branch-point) we have, if k be the
value of lu there,
where P [-) may, in general, be assumed not to vanish for z = cc ; so that
1 - - 1 -
- P {{z - y) '"I '" + - (z - jY" P' {{z - ryy
z- \z I z" \z ,
and therefore (^ (2^) has a zero of the second order at z = cc ,
Near a branch-point 7, where m sheets of the surface are connected, we
have
IV {z) - w (7) = {z- 7)'" P {{z - 7)'»|,
where P may, in general, be assumed not to vanish for ^ = 7 : hence
4> (^) = (^ - 7)
so that <^ (ir) is infinite at 2; = 7, and the infinity is of order ni — 1.
Hence the total number of infinities is 2 (m — 1), where m is the number
of sheets connected at a branch-point, and the summation extends over all
the r branch-points. But 2p -f- 1 = S (m — 1) — 2?i -|- 3, and therefore the
number of infinities is 2p — 2 -r 2n.
• We can now prove that the number of zeros of (f) (z) in the finite part
of the surface is 2p - 2, of tuhich p — I can be arbitrarilij assigned.
The total number of zeros is 2p — 2-\-2n, being equal to the number
of infinities because (^{z) is an algebraic function. But (^{z) has been
526 ALGEBRAIC FUNCTIONS [241.
proved to have a zero of the second order when ^ = qo ; and this occurs in
each of the n sheets, so that 2?i (and no more) of the infinities of (p (z)
are given by z=oo . There thus remain 2p — 2 zeros, distributed in the
finite part of the surface.
Moreover, the most general function (f) (z) of the present kind is of the
form
(z) = C, (/)i (z) +C,(f>,{z)+... + Cp(f>p (z),
where <^i (z), ..., (f)p (z) are derived through the normal functions of the first
kind. The p — 1 ratios of the constants C can be chosen so as to make
(b (z) vanish for p — 1 arbitrarily assigned points. Hence an algebraic
function arising as the derivative of an integral of the first kind is determined,
save as to a constant factor, by the assignment of p)—l of its zeros in the
finite part of the plane.
Note*. It may happen that the assumptions as to the forms of the series
in the vicinity of a particular point a, of oo , and of 7, are not justified.
If (/) (a) vanish, we may regard a as one of the 2p) — 2 zeros.
If 5 = 00 on one sheet be a zero of «^ {z) of order higher than two, say
2 + s, we may consider that s of the 2p — 2 zeros are removed from the
finite part of the surface to coincide with ^ = 00 .
j^
If P [{z — 7)"*} vanish for 2- = 7, the order of the infinity for ^ {z) is
reduced from m — 1 to, say, m — s—1; we may then consider that s of the
2p — 2 zeros coincide with the branch-point.
242. When the integer ^ of § 240 is greater than zero, so that a
rational function having m assigned simple poles can be expressed as a
linear combination of m — p + q functions each possessing only p -\-l—q
poles, then the rational function is called f a special function, to distinguish
it from the most general case, when g = and the points c are quite
arbitrary. In the case of a special function, having Ci, ..., c,,,, for its (simple)
poles, these points are such that q of the functions ^, say
^1, 02, ..., (^q,
vanish at each of them. Now in § 241 it was proved that the number of
zeros of any function for finite values of ^ is 2p—2; consequently, in
the case of a special function,
m^2p — 2,
or the degree of a special function is not greater than 2p — 2. Moreover,
q denotes the number of distinct adjoint curves of order n — 3, which pass
through the poles of the special function.
* See Klein-Fiicke, vol. i, p. 545.
t Klein-Fricke, Vorl. ii. d. Th. d. ell. Modulfunctionen, t. i, p. 552.
242.] SPECIAL ALGEBRAIC FUNCTIONS 527
Denoting the special function by g, and selecting any one of the q
adjoint polynomials of order n — 3, which occur in the q functions <^, say
U-i^{w, z), consider the product gU^^. This product is finite at each of the
points Ci, ..., Cm, because those points are zeros of U^ ; and g is not elsewhere
infinite on the surface. Consequently the integral
dw
is finite everywhere on the surface, and it is therefore an integral of the
first kind, say
[ V,dz
9/" ' -
dw
where Fj is the appropriate adjoint polynomial of order w — 3. Thus
or a special function is expressible as the quotient of one adjoint polynomial
of order n — S by another*.
Consider, in particular, the function
_ a^cf), + Of202 + . . ■ + aq({)q
where a-i^,\..,o.q are arbitrary constants. Each of the quantities (pi,...,(f>q
vanishes at the points Ci,...,Cni- The only infinities of gi are the zeros
of (f)i, which are only 2^ — 2 for finite values of z; and m of these 2^—2
zeros are not infinities of g^ , so that gi has only 2p — 2 — m infinities, say
where m =2p — 2 — m. Now, by the Riemann-Roch theorem of | 240, the
most general rational function, which has m simple poles (and no athers)
on the surface, contains m' —p + q' + l arbitrary constants, where q' is the
number of the functions
(or the number of linear combinations of them), which vanish at all the
points kx, k2, ...,k\n'. The arbitrary constants in gi are a^, ..., aq, being
q in number ; hence • »
q^m' —p + q + 1.
Now treat the m' points k-^^, ...,km' in the same way as the m points
Ci, ...,c-m have been treated; and let the analogous function be constructed.
Let (or linear combinations of them) vanish ; and
it contains q constants, so that, by another application of the Riemann-Roch
theorem, we have
q ^m — p + q + 1.
But m + m = 2p — 2, so that m — p + 1 + m' - p + 1 = 0; that is, the preceding
relations betAveen q and q' are equalities, so that
q = m' —p + q' + 1,
q' = m—p + q + l,
and therefore*
2(q — q') = m' — m,
which is called the Brill-Nother latu of reciprocity. It is a complement of
the Riemann-Roch theorem.
Since q is actually equal to m' —p + q +1, and does not merely possess
it for an upper limit, it follows that
a](^i + ... + aq^q
is a special function, which contains the largest admissible number of
arbitrary constants.
Note. The preceding investigations deal solely with those rational functions on a
Riemann's surface which have their poles of the first order. When we have to deal with
functions which have poles of order higher than unity, the investigations are much more
complicated and really belong to the general theory of Abelian functions. They will be
found, together with references, in Baker's Abelian Functiotis, ch. in.
The simplest result is contained in the following example.
£x. If a rational function is infinite at only a single point c on a Eiemann surface
the order of its infinity being m, and if the point c is perfectly arbitrary, then m must be
greater than p. (Weierstrass.)
243. The existence of functions that are uniform on the surface and,
except at points where they have assigned algebraical infinities, are finite
and continuous, has now been proved ; we proceed, as in § 99, to shew how
algebraical functions imply the existence of a fundamental equation, now to
be associated with the given surface.
The assigned algebraical infinities may be either at the branch -points,
or at ordinary points which are singularities only of the branch associated
with the sheet in which the ordinary points lie, or both at branch-points and
at ordinary points.
* Brill u. Nother, Math. Ann., t. vii, (1874), p. 283.
243.] FUNDAMENTAL EQUATION FOR THE SURFACE 529
Let the surface have n sheets; on the surface let the points Ci, Cg, •.■■,Cm,
be ordinary infinities of orders q^, q.y, ..., q^ respectively — we shall restrict
ourselves to the more special case in which g^, q^, ..., qm are finite integers,
thus excluding (merely for the present purpose) the case of isolated essential
singularities; and let the branch-points a^, a^,, ... be of orders pi, p^, ... as
infinities* and of orders n — 1, r^ — l, ... as winding-points.
Let w-i, W2, ..., Wn be the n values of the function for one and the same
arithmetical value of z ; and consider the function {w — w^) {w — w^ ...{tu — Wn).
The coefficients of w are symmetric functions of the values Wi, ..., w^ of the
assigned function.
An ordinary point for all the branches w is an ordinary point for each of
the coefficients.
An ordinary singularity of order q for any branch, which can occur only
for one branch, is an ordinary singularity of the same order for each of the
symmetric functions ; and therefore, merely on the score of all the ordinary
singularities, each of these symmetric functions can be expressed as a mero-
morphic function the denominator of which is the same polynomial function
m
of degree 2 q^ in z.
In the vicinity of the branch-point cv^, there are r-^ branches obtained from
{z-a,) ^^P{{z-a,)%
1
where P is finite when z = a^ by assigning to (z — a^)^' its r^ various values.
Then, as in § 99, the point a^ is no longer a branch-point of any of the
symmetric functions ; for some of the symmetric functions the point tti
is an accidental singularity of order p^, but for no one of tjiem is it a
singularity of higher order. Hence, merely on the score of the infinities at
branch -points, each of the symmetric functions can be expressed as a mero-
morphic function the denominator of which is the same polynomial function
of degree 2_pi in ^.
No other points on the surface need be taken into account. If, then, P (z)
be the denominator of the coefficients arising through the isolated algebraical
TO
singularities, so that P (z) is of degree S qs in z, and if Q (z) be the de-
.9=1
nominator of the coefficients arising through the infinities at the branch-
points, then
P (z) Q (z) (tU - Wi) (W - tU^) ...{lU - Wn)
* A branch-point a is said to be an infinity of order p and a winding-point of order r-1,
_2 1
when the affected branches in its Yicinity can be expressed in the form {z-a) ''P {(^-a)'"},
where P is finite when z = a.
F. F. . - 34
530 FUNDAMENTAL EQUATION [243.
is a rational integral function of w and z', say f{w, z), which is evidently
m
of degree n in w and of degree "^ qs-\- Xp in z.
s=l
Its only roots are iu = Wi, ..., Wn', that is, the function w on the Riemann's
surfece is determined as the root of the equation f(w, z) = 0; and therefore
the equation f(w, z) = is a fundamental equation, to be associated with
the surface.
Ex. 1. Shew that a fundamental equation for a three-sheeted surface, having e"^"*'^^ (for
i {z), where
4)i {z) = e" (^^ ,
is uniform on the resolved Riemann's surface : it has a single zero (of the first
order) at (3-1 and a single accidental singularity (of the first order) at «! ; its
factor for the cross-cut a^ is unity, and its factor for the cross-cut hr is
g2{i«r(|8i)-Wr(ai)}
The function i {z) may therefore be regarded as an element for the repre-
sentation of a factorial function.
Let ^ {z) be a factorial function on the Riemann's surface with given
multipliers m and n; and let it have a number q of zeros /Sj, ^^, ..., /3g, each
of the first order, and the same number q of simple accidental singularities
«!, 0L2, ..., ofg, each of the first order, and no others. Then ^' {z)l^{z) has 2g'
accidental singularities ; in the vicinity of the q points jB, it is of the form
and in the vicinity of the q points a, it is of the form
1
z— a
+ P(z-a);
^'(z) ^
hence ___2^^/(.)
* It may be pointed out that this result is an illustration of the remark, at the beginning of
§ 244, that the factorial functions have a uniform function of position on the surface for their
argument and not the integrals of the first kind, of which that variable of position is a multiply-
periodic function.
534 EXPEESSIONS FOE [244.
is finite in the vicinity of all the singularities of ^ . . Thus
s = l
has no logarithmic infinities on the surface : neither log (z) nor any one
of the functions tt (z) has infinities of any other kind ; and therefore the
foregoing function is finite everywhere on the surface. It is thus an integral
of the first kind ; so it is expressible in the form
2Xi Wi (0) + 2X2^/2 (2) + ... + 2\pWp (z) + constant.
Hence (^(z) = Ae'=' '^=^
where J. is a constant.
The function represented by the right-hand side evidently has the q
points /S as simple zeros and the g points a as simple accidental infinities,
and no others. Higher order of a zero or an infinity is permitted by repeti-
tions in the respective assigned series.
In order that it may acquire the factor m^ on passing from the negative
edge to the positive edge of the cross-cut a,., we have
and that it may acquire the factor n^ in passing from the negative edge to
the positive edge of the cross-cut br, Ave have
a p
The former equations determine the constants X,,. in the form
1 ,
\r=Tr--^ogmr,
for r = 1, 2, ..., p ; the latter equations then give
q 1 ^
X [Wr (/3,) - Wr (Ot,)} = i log tl^ " 9— ■ ^ (Bjcr log m^),
S=l ■"'''■ k = l
for r = 1, 2, ..., p.
Apparently, X^ is determinate save as to an additive integer, say 31^ ; and
the value of -^log?v is determinate save as to an additive quantity, say Nriri,
where JS^r is an integer. The left-hand side of the derived set of equations
being definite, these integers JSf^ and Mr must be subject to the equations
k=l
244.] FACTORIAL FUNCTIONS 536
for r= 1, 2, ..., p; and therefore, equating the real parts (§ 235), we have
I M^^pkr = 0,
A: = l
SO that ^ 2 MkMrpkr = 0,
k = l r=l
which, by § 235, can be satisfied only if all the integers Mr vanish and there-
fore also the integers Nr.
Hence when the foregoing equations connecting the quantities a, 13, log n,
log m are satisfied, as they must be, for one set of values of log n and log m,
that set may be taken as the definite set of values ; and the only way in
which variation can enter is through the multiplicity in value of the functions
Wi, ..,, Wp, which may be supposed definitely assigned.
The expression for the function O {z) is therefore
7 1 P
2 7rs(z)+—, 2 {wAz)iosm.}
Ae'='
the q zeros /9 and the q simple poles a being subject to the equations
S \Wr (/3,) - w, (a^)} = \ log ??,. - ^^— ^ 2 {B^r log m^fc).
Corollary I. The function $ {z) is a rational function of position on
the surface, that is, of w and z, if all the factors n and m be unity. Such a
function has been proved (§ 194) to have as many infinities as zeros; and
therefore integers N-l, ... , Np , i//, ..., MJ exist such that, between the zeros and
the infinities of a rational function of w and z, the p equations
2 [Wr (^s) - Wr («,)} = iriN; - I M^Bkr,
s=l k=\
for r = 1, 2, ..., p, subsist*.
The function O (z) then corresponds to a rational function, when regarded
as a product of simple factors, in the same way as the expression (§ 241)
in terms of normal elementary functions of the second kind corresponds
to the function, when regarded as a sum of simple fractions.
Corollary II. Every factorial function has as many zeros as it has
infinities.
For if a special function <^ (z), with the given factors and possessing q zeros
and q infinities, be formed, every other function with those factors is included
in the form
F{z)=^(z)R(w, z),
where R (w, z) is a rational function of w and z. But R iw, z) has as many
zeros as it has infinities ; and therefore the property holds of F {z).
* Neumann, p. 275.
536 FACTORIAL FUNCTIONS [244.
Further, it is easy to see that the equations of relation between the zeros,
the infinities and the multipliers are satisfied for F{z). For among the zeros
and the infinities of '^■{z), the relations
q 1 J'
S [Wr (/3s) - Wr (tts)} = i log llr - 9— • S {B^r log m,,)
S = l '^^'' Jc = l
are satisfied ; and among the zeros and the infinities of R (w, z), the relations
S Wr (A') - Wr (a/) = iriN; - I {Bi,r M„')
s=l it=l
are satisfied, where i\V and the coefficients M' are integers. Hence, among
the zeros and the infinities of F (z), the relations
1 ^
t \wr (zero) -Wr{cc)] = ^ (log Ur + Nr 27ri) - K— • 2 [Bj,r (log mjfc + 2 Wtti)}
are satisfied, giving the same multipliers n^ and w^ as for the special function
(4
Corollary III. It is possible to have factorial functions without zeros
and therefore without infinities: hut the multipliers cannot he arbitrarily
assigned.
Such a function is evidently given by ^
derived from {z) by dropping from the exponential the terms dependent
upon the functions tt (2;). The relations between the factors are easily
obtained.
Note. The effect of the p relations
q 1 *^ •
2 [Wr (/3s) - Wr (Gs)] = h log ?^r " ^T"" ^ (Bkr log mfc),
subsisting between the factors, the zeros and the infinities of the factorial
function, varies according to the magnitude of q.
If q be equal to or be greater than p, it is evident that all the infinities a
and ^— j3 of the zeros ^ can be assumed at will and that the above relations
determine the p remaining zeros. The function therefore involves 2q—p
arbitrary elements, in addition to the unessential constant A.
In particular, when q is equal to p, the infinities a can be chosen at will
and the zeros /3 are then determined by the relations. It therefore appears
that a factorial function, which has only p infinities, is determined hy its
infinities and its cross-cut factors.
When q is greater than p, say = p+ r, then the q infinities and r zeros
may be chosen at will. By assigning various sets of r zeros with a given set
of infinities, various functions ^1 (z), ^.^ (z), . . . will be obtained all having the
244.] BIRATIONAL TRANSFORMATION 537
same infinities and the same cross-cut factors. Let s such functions have
been obtained ; consider the function
it will evidently have the assigned infinities and the assigned cross-cut
factors. Then s—1 ratios of the quantities /jl can be chosen so as to cause
(z) to acquire s — 1 arbitrary zeros. The greatest number of arbitrary
zeros that can be assigned to a function is r, which is therefore the greatest
value of s — 1. Hence it follows that ?' + 1 linearly independent factorial
functions i {z), ..., ^r+i (2) exist, having assigned cross-cut factors and p + r
assigned infinities ; and every other factorial function with those infinities and
cross-cut factors can he expressed in the form
/^i^i {z) + ^lf^■2 {z) + ...+ /i,-+i^r+i {z),
tuhere /jl^, ..., /x^+i are constants whose ratios can he tised to assign r arhitrary
zeros to the function.
These factorial functions are used by Appell to construct new classes of functions in a
manner similar to that in which Riemann constructs the Abelian transcendents. Their
properties are developed on the basis of algebraic functions ; but as only the introduction
to the theory can be given here, recoiirse must be had to Appell's interesting memoir,
already cited. See also Baker's Abelian Functions, ch. xiv.
BiRATioNAL Transformation.
245. It has already been pointed out (§ 193) that, if w' denote any
arbitrary rational function on a Riemann's surface (say S) associated with the
relation f{iv, z) = 0, then w' satisfies an equation f (w', z) = 0. Similarly,
if / denote another rational function on S, the elimination of w and z
between the three algebraical equations
w' = R^ (w, z), z = R2 (w, z), f(w, z) = 0,
leads to another equation F {iv , z) = 0.
Relations such as these, which express new variables w' and z' as rational
functions of old variables w and z, are called transformations: sometimes
rational transformations. Transformations exist between two sets of variables,
each set being regarded as a pair of independent variables: with such
transformations (which include the well-known Cremona transformations) we
are not specially concerned, seeing that our variables w and z are connected
by a permanent equation f{tu,z) = 0. We have to deal* with rational
transformations between two equations such as
f{w,z) = 0, F(w',z') = 0.
* The difference is the same as the difference between the rational transformations of a plane
and the rational transformations of a curve in the plane; the former give rise to the latter,
though not to the whole of the latter.
538 TRANSFORMATION OF . [245.
The degree of F in w' and /, when the orders of w' and z' are known, can
be inferred. Suppose that /= is of degree m in lu and of degree nm z;
and let the orders of tv' and z', defined by the equations
w' = Ri (lu, z), z = i^o {Wy z),
be m' and ?i' respectively. There are m positions on S which correspond to
any given value of w' ; each such position gives one value of z' ; and therefore
there are m values of z' for any given value of w'. Similarly, there are 7i'
values of w' for any given value of /. Accordingly, the equation
F(tu',z') =
is of degree n' in w', and of degree w' in /.
As the two rational functions of position on S, represented by w' and z',
are quite unrestricted, it follows that we can obtain an unlimited number of
transformations of the equation f=0. We assume that /' is an irreducible
polynomial, that is, ^cannot be resolved into factors rational in w and z, so that
/= is an irreducible equation ; and we find that the equation F = 0, arising
out of any rational transformation of /=0, is such that the polynomial F
either is irreducible or is some poiver of an irreducible polynomial. For let
TTq and Z^ denote any values of w and z, which satisfy ^ = ; they arise
through some position iv^, Zq on S: and let Wj and Z^^ denote any other
values of w' and z, which satisfy F = 0; they arise through some position Wj,
Zi on S. (In each case, there may be more than one position.) We can pass
from Wq, Zo to w^, z^ on ^ by a continuous path, which avoids all the branch-
poirlts, and which does not pass through any infinity of w' or z' ; during the
passage the values Wq and Z^ change continuously into W^ and Z^^. Hence
when we have constructed the Riemann's surface (say S') associated with
F =0, and take account of the dependence of w' and / upon w and z that
leads to F=0, it follows that, on this new surface S', a continuous path exists
which joins the position Wq, Z^ to the position TTj, Z^. Also these positions
are any positions on 8', because the values W^, Z,^', Wj, Z^; are any values
that satisfy ^ = 0; hence (§ 176, Ex. 5, Cor. II.) F either is an irreducible
polynomial or, if reducible, is some power of an irreducible polynomial.
Consider any position w', z' on -8'. To the value of iv , there correspond
m! positions on 8, say
and to the value of /, there correspond ri positions on *S^, say
/Si, 61; A> ^2; •••; Ai'> ^ft'-
Then as w\ z constitute a position on 8\ it follows that one (or more than
one) position on 8 must be common to the two sets. First, let only one
position be common to the two sets. In that case, the simultaneous values
of w' and / (which determine a position on >S") determine a single position on
245.] ALGEBRAIC EQUATIONS 539
S, that is, give w and z uniquely. Now w and z, as functions of position
on S', at"e manifestly not transcendental; it has just been proved that they
are uniquely determined by iv' and /; and they consequently are rational
functions of position on 8', that is, we have
lu = S, (w', /), z = S, {w\ /), F (w', z') = 0,
where S-^ and S2 are rational functions. Moreover in this case, to a general
value of z', there correspond n' different positions on S; each of these
determines a value of tv', so that there are n values of lu' ; these values are
all different, for taking the single value of z and the various values of w'
in turn, the n' positions in the second set must be exhausted, and no first
set has more than one point common with the second set. Hence to a
value of /, there correspond n different values of w'. Also F is either an
irreducible polynomial of degree n' in tu', or it is a power of some irreducible
polynomial ; in the present case, therefore, F is. irreducible. Accordingly,
the surfaces 8 and 8' associated with the two equations
f{w,z) = 0, F(iv',z') =
are such that each position on one determines one (and only one) position
on the other ; and the variables of each position are expressible rationally in
terms of the other, in forms
w' = -Ri (w, z)\ w = 8-^ (w', z')]
z' = R2 (w, z)} ■ z = 82 (w', z')\
where H^, Ro, 8^, 8.2 are rational functions. Such a transformation is called
birational.
Next, let I of the positions on 8 be common to the two sets, which give
the values of w' and / respectively. To the position a^, a^ in the w'-set,
there corresponds a definite value of z' ; as I positions on 8 arise through
given values of w' and z', it follows that other I — 1 positions in the ly'-set
give the same value of /. Let these be Cg, ag; ... ; oli, ai; so that no other
position in the w'-set gives that value of /. Take now some other position
a^+i, a^+i; it gives a definite value to /, and there are other ^— 1 positions,
say a;_,.2, a^^-a ; ...; a^i, aazV which give that value. Proceeding in this way,
we see that m must be a multiple of I, say in' = 'ni"l ; and that the one value
of w , which gives m' positions on 8, gives rise to m" values of /, each
of them repeated I times. Dealing similarly with the ^'-set, we see that n
is a multiple of I, say n = n"l ; and that the one value of /, which gives
n' positions on 8, gives rise to n" values of w , each of them repeated I times.
In this case, F is the lih. power of an irreducible polynomial F^ {w', z'), of
degree n" in w and degree m" in / ; and the surfaces 8 and 8' associated
with the two equations
f{w,z) = 0, F,(tv,z') =
are such that to one position on 8, there corresponds only a single position
on 8' : while to one position on ;S^', there correspond I positions on 8. The
540 RATIONAL TRANSFORMATIONS [245.
transformation from f to F^ is rational ; it is not rational fi^om F^ to /; that
is, the transformation is not birational.
Sometimes the results are expressed in geometrical language by saying
that, in the former case, there is a (1, 1) correspondence between the curves
/= 0, F = 0: and in the latter case a (1, I) correspondence between the
curves /= 0, Fi = 0. The whole subject of rational transformation is involved
in the theory of correspondence between curves.
JS^ote 1. It has been proved that, in the equation F(w', z) = obtained
by eliminating w and z between
w' = i^i (w, z), z' = R.2 {iv, z), f{w, z) — 0,
the polynomial F either is irreducible or it is some power of an irreducible
polynomial. Now when the values iv' = R^, z' = R.j, are substituted in F—0,
the result is to give an equation in w, z only; so that, F(Ri, R^) must have
f{iv, z) as a factor. It is not possible to prove that i^ is a power of /(w, z),
because this is not always the case. As one example of this remark, let
tv = T,{W,Z), z=T,(W,Z),
be substitutions, which leave w' and z' unchanged in form, that is, give
vj' = R, ( W, Z), z'=R,{ W, Z) ;
the equation F (w', z') = will be obtained in association with the relation
/(Ti, Tz) = 0; and therefore F will contain f{T-^{w, z), T^iw, z)] as a factor.
Thus when we substitute for w' and / in F, the resulting expression may be
divisible by factors other than f{iu, z).
Note 2. When ^ is a power of a pol3rnomial, some special process of the
elimination indicated on p. 537 may lead, not to F, but immediately to the
polynomial. The explanation is that the eliminant then obtained is not of
the proper degree in w and z' as required on p. 538. As a trivial example,
we see that the equations
w^ + z^ = 1, w' = w^, z' = z^.
lead at once to w' + z' — \, whereas F{w', z) is {w' + z' — ly.
Ex. 1. Consider the transformation of the equation ^^ + ^^ = 1 by the relations
Jcvf = wz, %z' = aw + cz.
As regards the degrees of vJ and 2', each of these variables is infinite only when \z\^
and so | w|, is infinite. There are three such positions on the surface; at each of them,
w' is infinite of the second order, and z of the first order ; so that wi', the degree of w\
is 6 ; and w', the degree of z\ is 3. Accordingly, the equation between w and 0' must be
of degree 3 in to' and degree 6 in z .
We have
{aw — cs)^ = 4 (s'^ - kaciv') = iA^,
say ; so that
aw = z' + di., cz = z' — ^.
245.] EXAMPLES 541
Substituting in w^ + z^=l, we have
= (c^ + a3) {4z''^ — 3kaciv') z' + (c^- a^) A {'iz'^ — kacw'),
and therefore
(c3 _ a3)2 {iz'^-kacivy (s'^ - kacio') - {a^c^ - {c^ + a^) (4s'2 - Zkacw') /}2=0,
which is the equation F{;w', z') = 0. Manifestly i^is irreducible, so that the transformation
is birational ; in fact,
, a^c^ — { .(where F is of degree n in lu'), which belong to the p normal integrals
of F of the first kind. We have
■rrr dz, Vk =
div
dF '
dw'
so that the above p relations between u and v give p differential relations of
the form
■rpdz = ^(c^jV, + c^2V2+ ...+c^pVp),
dw div'
544 EQUATIONS OF BIKATIONAL TRANSFORMATION [246.
satisfied in virtue of the birational transformation. Hence, when p>l,we
have
or the ratio of two adjoint polynomials of order ?i — 3 for the one surface is
transformed into the ratio of two adjoint polynomials of order n — 8 for the
other surface, when the transformation is birational. When p = 1, we merely
have
U . V ,
dj_ dF
dw dw'
satisfied in connection with the birational transformation ; and when p = 0,
there is no relation.
The transformations of equations of genus or 1 are to be considered
separately. It is clear that, when two equations of the same genus greater
than unity, are known to be birationally transformable into one another, the
equations
Ui CiiFi + ... + CipFp
can be used to obtain the birational transformation.
As a birational transformation conserves the genus of the equation to
which it is applied, we naturally regard all equations, which are birationally
transformable into one another, as belonging to the same class : and we- have
to determine what are the characteristics other than conserved genus upon
which the class depends.
To obtain these, take an equation of genus p(> 1) ; (equations of genus
and 1 will be considered separately, from this point of view as well as for
the reason above) : and on the Riemann's surface of the equation, take a
rational function /, having /j, poles each of the first order. Let the positions
of the /Lt poles be chosen quite arbitrarily, and let their number be >2p — 2,
so that the rational function is not a special function (§ 242). Now / contains
fi- p + 1 arbitrary constants, which enter linearly (| 240 : the number q
is zero, because /jb>2p — 2): and therefore as the positions of the poles are
arbitrary, each of them accordingly being determined by an arbitrary quantity,
it follows that the total number of arbitrary constants in z' is
(fi -p + l) + fj., = 2fi -p + 1.
Choose z' as the independent variable for a transformed equation. Since
the degree of / on the original surface is /x, being the number of its
infinities, we know (| 245) that the degree of the new equation in its
246.] CLASS-MODULI 545
dependent variable is equal to ix ; hence the new Riemann's surface is
/i-sheeted. As its genus is equal to p, its ramification is given by
n = 2(/^+i)-l).
Now when the branch-points and the branchings of a surface, of given genus
and given number of sheets, are assigned, the surface is definitely known as
(at the utmost) one of a limited number (§ 212 : footnote). The corresponding
equation is then known (§ 193) so that, as z is known, the dependent variable
can be regarded as determined by the assignment of the ramification : it
contains no independent arbitrary element.
We have 2^ — _p + 1 disposable constants by which to meet the demands
of the ramification, which amount to 2/Li 4- 2j9 — 2 constants : hence there are
'ip - 3,
= 2yu, 4- 2|) — 2 — (2/x — j!> -I- 1), constants surviving, as undetermined by the
arbitrary elements in z'. The transformation is, of course, definite ; and
therefore these Zp — 3 quantities are determined by the first surface.
It therefore follows* that the class of equations, which are birationally
transformable into one another, are determined by 3jw — 3 quantities ; they
are called the class-moduli of the equations.
In this result, which is due to Riemann, one modification must be made,
as pointed out by Klein f. In the course of the proof, it was assumed that
all the 2fj,—p + l disposable constants could be used to determine 2yu, — p -f 1
quantities connected with the ramification. As will be seen, a surface may
be transformable into itself by a birational transformation; and it might
happen, in such a case, that the transformation contained arbitrary constants.
If p be the number of these arbitrary constants, then it follows that, in the
transformation under our earlier consideration, we cannot use more than
2/uL — p + 1 — p of the arbitrary constants in / for the ramification of the
surface ; and therefore the number of class-moduli is
n - (2^ -p + l-p)
= Sp — S + p.
As a matter of fact, p = when j9 > 1 (§ 250) ; p = 1 when p = 1 (§ 248) ;
p =^S when p = (§ 247) ; all of which results will be established later.
Ex. 1. Consider the equations
where Z^ is a sextic function of 2, «4 is a quartic and % is a quadratic in i'. Each of the
equations is of genus 2 ; and therefore it may be expected that, if they are birationally
* Eiemann, Ges. Werke, p. 113. It is assumed thioughout that each equation is comijletely
general : it may happen that, for equations which are special in form, the number of class-moduli
is less than dp - 3.
t Ueher Riemann's Tluorie der algehraischen Functionen, (Leipzig, Teubner, 1882], p. 65.
F. F. 35
546 EXAMPLES OF [246.
transformable into one another, three ( = 3.2-3) relations among their constants will
be satisfied.
Integrals of the first kind belonging to the first equation are
I Z^ 2 dz^ I zZq ^ dz ;
and integrals of the first kind belonging to the second are
I (Mo %) ~ ^ dz', j Z {U2 Ui) " 2 dz'.
As the equations are to be rationally transformable into one another, we have (§ 246)
Zq~^ dz — y {U2tii)~ ^ dz' -\- 8z' (2t2%)~''^ d^-,
zZe~^ dz = a (u2Ui)~^ dz +^z' {u^Ui)"^ dz',
and therefore
a + ^z'
y + ^z'
Take w = iv'u2K,
whei'e A' is some function of z ; then
ZQ — K^u^Ui.
In other words, the effect of substituting {a + ^z')l{y+bz') for z in the sextic Z^ must be to
give a multiple of the sextic U2Ui ; so that, taking
where e is a constant, we have
ZQ{a+l:iz', y + S2') = e"^«2«4.
In order that one sextic may be transformable into another by a substitution
/ : l=a + /3/ : y + bz',
they must have their invariants the same save as to a factor. The invariants of a sextic
are of deo-ree 2, 4, 6, 10 (as well as one of degree 1.5, the square of which is expressible as
an integral function of the others); denoting them for Z^ by Li, I^, !&, Jw, and for
^2% ^^y ^^2) Ji^ Jqi '-^w respectively, three relations as required are
7^/2-2 =J^J,^-'^]
In order to find the actual transformations, we compare the coefficients in
ZQ{a+^z', y+8z') = eh(,2th-
There are seven equations, each expressing some homogeneous combination of dimensions
six in a, /3, y, S in terms of e'-^ and constants. The equations are equivalent to four
in virtue of the preceding three relations ; they therefore suffice for the determination
of a, Idj y, S. The transformation thus is
a + 3z' eU2
y+6z" {y + 8z'f
which happens to be a Cremona transformation.
Ex. 2. Consider the birational transformations of the equation
where U is a quartic function of z.
246.] BIRATIONAL TRANSFORMATION 547
Let H, $ denote the Hessian and the cubicovariant of U respectively : /, J its quadrin-
variant and its cubinvariant. Then
20.= ^^- 6^^
^z dz'
$2 3
m^^ium-
-JU^.
Take a new va]
fiable z'
such
that
z'U+H=Q;
then
dz' 2$
dz~ U^-'
and therefore
9,
dz
dz'
Ui ' (4£'3-Iz'-J)h'
Accordingly, the integral of the first kind, belonging to the equation
becomes an integral of the first kind, belonging to the equation
the equations of (birational) transformation being
U , $
The integral of the first kind can be transformed slightly by writing
z'I=JC;
it becomes
^ (4K'-C-i)*
where
A denoting the discriminant of the original quartic, viz. A = P — 27J^. Thus
2/ dz _ dC
so that the integral of the first kind associated with the original equation becomes
a constant multiple of the integral of the first kind belonging to the equation
with which it is birationally related.
In order to determine all the equations of the same class as w"^ — U= 0, it is clear that,
in each case, their integral of the first kind must be a constant multiple of
dc
that is, the constant p is the (sole) class-modulus. It manifestly is the single absolute
invariant possessed by the quartic ; and so we infer the result that the equations
w'^=U{z,\\ w'2=F(/, 1),
ivhere U and V are quartic fimctions of z and ^, are birationally transformable into
one another, if the quartics have equal absolute invariants*.
* Hermite, Crelle, t. lii, (1856), p. 8.
35—2
548 EQUATIONS OF [246.
In particular, consider the equation
if it is to belong to the same class as iv^= U {z, 1), its absolute invariant must be the same.
Now
accordingly, the condition is that c satisfies the equation
108c (1-c)* ~ A"
As a A^ery special case, we infer that the equations
are birationally trans foi'mable into one another if
(H-14a + a2)3_(l + 14c + c2)3
a{l-af c(l-c)* ■
The actual construction of the transformations is left as an exercise.
Ex. 3. Obtain Riemann's theorem as to the number of class-moduli of a class of
algebraic equations from a relation of the type
V = aiUi+ +ap2i (/^) ' (p2 in) '
and denote by ^ (/x) the greatest common measure of cpi (/x) and (^2 (/"•)• Now
any straight line, say
Aiu + Bz + C = 0,
cuts the curve /= in n points ; and therefore the equation
must give n values of /x, one for each point, that is, it must be of degree n.
Hence the degrees of
cannot be greater than n.
* This result is in accordance with § 205. For U is an adjoint polynomial ; the number of
adjoint polynomials, which are of degree 71 - 3, isp, viz., zero in the present case,
550 EQUATIONS OF GENUS ZERO [247.
Conversely, if the variables w and z of an equation f=0 are rationally
expressible in terms of an arbitrary parameter, the equation is of genus zero.
For if the genus were greater than zero, an integral of the first kind, say
dw
would exist which would be finite everywhere on the associated Riemann's
surface. Substituting for w and z their values in terms of ft, we should have
the integral
R (/ju) d/ji
(where E is a rational function) finite for all values of /n — an impossible
result. Hence the genus of the equation must be zero.
Further, any curve (or equation) of genus zero can be birationally trans-
formed into any other curve {or equation) of genus zero by relations which
involve three arbitrary parameters. Let one of the equations be represented
by
w = Ri ifM), z = R2 ilj), IJ^ = R {w, z) ;
and the other by
w' = &^{\), / = ^2(^), \ = ^{w',z');
where R-^, R2, S-^, Sq, R, S are rational fanctions. In a birational transforma-
tion, one set of values of w and z determines one set of values of w' and z',
and vice versa ; therefore one value of fi determines one of A,, and vice versa,
so that the relation between \ and /i, is of the form
_a\ + b
^~ c\ + d'
where a, b, c, d are arbitrary. This relation, containing the three arbitrary
parameters a : b : c : d, gives the birational transformation
„ faS +b\ r> fci^S + b
^cR — aJ' ^\cR — aj
establishing the proposition.
It is an immediate corollary that any curve of genus zero can be biration-
ally transformed into itself by equations that contain three arbitrxiry parameters ;
thus the quantity p of p. 545 is 3 when p = i). If desired, the three parameters
can be determined so that any three assigned points correspond to three other
assigned points.
247.] SUB-RATIONAL REPRESENTATION 551
Note. It may happen, in a particular instance, that the actual expressions
for w and z in terms of the parameter are obtained in a different manner, so
that
w = R, (X), z = Ro^ (X),
but that \ is not a rational function of tu and z. Thus the possibility could
arise from the preceding result by taking
/jb = S (A,),
where *S is a rational function of X not of the form j : we then should
cX + a
only have SCX) equal to a rational function of z. Such a representation,
which may be called sub-rational, is easily detected in fact, because the
equation
Aw + Bz+ C=0
then gives more than n values of A, ; and it can be corrected in form by the
suitable inverse substitution, which can be obtained as follows*.
Suppose that, in the expressions
0, (X) e. (X)
w = — --^ , z = , . ,
where the quantities 6^, 0.,, ^^i, -^o are polynomials, s values of A,, say Xj, X2, . . ., Xs,
correspond to given values of lu and z: then the equations
E, (A) = d, (X) X, M - 0, (AO Xl (^) = 0,
E, (A) = 6, (A) X. M - e, (A,) X. (^) - 0,
have the s roots A = Aj, Aj, ..., Xg common. Also each of these roots is simple
for each of these equations ; because if any one were multiple, say, Aj for
E^ (A) = 0, then it would satisfy
^/(A-)Xi(A0-^x(A0%/(A) = O,
so that
^/(A)^ ^/(A)
when A = Aj. The quantity Aj would then satisfy an algebraic equation
the coefficients of which are non-parametric constants — a result obviously
excluded when Ai (and so the other values of Ao, ..., A^) are parametric. We
therefore can obtain the greatest common measure of E-^ (A) and E2 (A) in the
form
(A - Aj) (A - As) ... (A - Xs)
= \' - /zi A'~' + ^l. X'-' - ....
Now not all the quantities /n can be absolute constants : some at any rate
must be a function of Aj, say /tx,. is such a function. But /i,. is a symmetric
* Liiroth, lilath. Ann., t. ix, (1876), pp. 163—165.
552 EQUATIONS OF [247.
function of Xj, ..., X^, so that it does not change its value when X,, ..., X^
are substituted for Xi ; hence it acquires only a single value for given values
of w and z. Moreover, to a given value of /li,- correspond s values of X ; if one
of these be Xi, the others are A,, ..., X,., because ft is a symmetric function of
Xj, ..., Xg. Hence to a given value of //.;., there correspond a single value
of w and a single value of z : that is, when the equations
_^i(X) ^ O2 (X)
are transformed by the relation
/LL = Hr (X),
the result is of the form
"^1 (/^) „ "^2 (fJ^)
w =
<^l (^) ' <^2 (^) '
z = -j-^ — - , fjb = R (iv, z).
Ex. 1. The equation*
(2/2 + Qxy + A'2)2 = IQxij {\yx - Zx -Zy + 4)2
is of genus ; so that x and y are rationally expressible in terms of a variable jDarameter.
To obtain their expressions, we notice that xy must be a perfect square ; so that, writing
xy = 6'^, .r+y = ^,
we have
and therefore
(^ + 6^)2 = 16^(1 + ^)2.
Hence ^ is a perfect square, say 6=\^; then
xy = \^, a; + 3/ = /x=4X— 6X24-4A^
Accordingly
(y_a;)2 = ;x2_4X4
= (4X - 8X2 4. 4X3) (4x _ 4X2 + 4X3)
= 16X2(1-X)2(1-X + X2),
P
so that 1 -X+X2 must be a perfect sqviare. Take A = 7^, so that P- - PQ + Q- is a perfect
square. This form will be secured for P'-PQ+Q'\ ={P+aQ) (P+co^Q), where « is
a cube root of unity, by writing
P + $co = (a-«2)2, P+(^co2 = (a-«)2,
SO that P=2a + a2, Q=l+2a, P^- - PQ + Q'- = {1 + a + a-y-. which gives
_ 2a+_a2 l-a 2 l+a + a2
^~'^~ Y+2^ ' T+2a' l+2a '
Also
^2a + a2 2 + 2a + a2 + 2a3 + 2a^
•^ '^ l+2a (l+2a)-
hence
/'2+a\3 , 2 + a
^ = nrT2a)' ^' = «l+2-a-
* It is one form of the modular equatiou in the cubic transformation in elliptic functions ;
Cayley, Coll. Math. Papers, t. ix, p. 170.
247.] GENUS ZERO ' 553
It is clear that the line
gives rise to six values of a ; it cuts the original unicursal sextic in six points ; and
therefore the expressions are rational, not merely sub-rational. To express a in terras
of X and y, let
_ y'^ + Qxy + x^' _
then
{ l + '2a j
so that
as may be verified directly by substitutuig the values of x and y. Also
y_ r 2 + a
^- ia(l+2a)j '
SO that
U X .
4y
= .r-l-2aA'-2a3,
from the value of x ; and therefore
Also
so that
accordingly
and therefore
^-iyr=2a(.r-l) + 2a(l-a2).
?7_ a(l+2a)
4j/~ 2 + a '
(^-l).^. = 2a(.^--l) + a(2 + a)(l
a2 + 2a ^ + A-=0.
From the expression for — , ^ve have
^«^+«"-|)-|-«-
Subtract this equation from twice the preceding quadratic ; and we have
^^ .
^ u V ^y)\ '2y
4j/
which, on substituting \'6xy for V^ when it occurs, leads to
4.ry-4A-+ U{l-y )
~~ dixy — 2y — ^x— U
554 EQUATIONS OF [247.
Ex. 2. Shew that the coordiDates of the curve in the preceding example can be
rationally expressed in terms of an arbitrary parameter jS, by taking
/3*- 682 + 4/3
4^3 - Qli'i + 1
as a value for X in the investigation. Obtain the relation between a and /3 : and thence
(or otherwise) shew that this representation is sub-rational.
Ex. 3. Discuss the curve represented by the equations
Ex. 4. Obtain relations of birational transformation which transform the unicursal
quartic
^2y2 _ 2_j.^ (^^ -p ^ fiy-^ _j. Qj_^2 _|_ 2bxy + cy"^ =
into the circle x^+y'^ = l.'
248. Some of the simpler properties possessed by equations (or curves)
of genus unity* can be obtained similarly. In the first place, we have
Clebsch's theorem that the variables can he expressed as rational functions
of a parameter 6, and of @^ where is a polynomial of either the third or
the fourth degree in 6. To establish this result, we take an adjoint polynomial
U of order n — linw and z, where / is of order n ; we make it vanish at each
of the multiple points of/ to the multiplicity A, — 1, when X, is the multiplicity
of the multiple point of /; and we make it pass through n — 2 arbitrarily
assigned points on /= 0. Then the number of remaining intersections of
U=0 and /= is
n (n - 2) - (n -2)-tX{\- 1).
But (§ 240) we have
'^\{\ - 1) = (n - 1) (n - 2) - 2p
= n (n - 3),
in this case ; and therefore the remaining number of points of intersection is
n (n - 2) - (n - 2) - n (n - 3), = 2.
Let C/^i = 0, U2 = 0, be any two curves satisfying all the conditions of U, as
regards its order, and its relations to the multiple points of /, and the n — 2
arbitrarily selected points on/; then
where 6 is arbitrary, is another such curve. It cuts/= in two points, other
than the multiple points and the assigned ?i — 2 points ; hence, eliminating z
between
u, + eu, = o, /=0,
and removing from the eliminant the factors that correspond to the multiple
points and the assigned points, the remaining factor must give the values of w
for the two points, that is, it is a quadratic in w. Hence we have
* For a full discussion, see Clebsch, Crelle, t. Isiv, (1865), pp. 210 270.
248.] GENUS UNITY 555
where A and B are rational functions of 6, and contains no repeated factor.
Similarly, by eliminating w, we should have z=^ C + D^^'^, where C and D are
rational functions of 0, and ©i contains no repeated factor. Substituting
these values of w and z in 11^+ OUz^O, /=0, the equations are to be
satisfied; and therefore the radicals @^, ©^^ are the same. Thus we have
Moreover
B '
the first represents ^ as a one-valued function of lu and z ; the second, on
substitution of this value for 0, represents ©^ as a one-valued function of w
and 2. Hence, writing
e = z', ©i = w',
we have ■
w'-'=%{e) = %{z)\
results which shew that the equations
/=0, iv'- = %{z),
are birationally related by the equations
10 = A {z') + %v'B {z'\ z=C {z') -f- iv'D (/).
Now when two equations are birationally related, we know (§ 246) that their
genus is the same ; hence the genus of w'^ = © (/) must be unity. This can
be the case only if © (z) is a cubic or a quartic polynomial in z ; and there-
fore @ is a polynomial of either the third or the fourth degree in 6. The
proposition is established.
Such curves (or equations) are called bicursal by Cayley*; they also are
sometimes called elliptic, because the equation w''^ = © (/) is associated with
elliptic functions, an association that leads to another mode of expression,
as follows. If © be of the third degree in d, a linear transformation of the
form ^ = a + h6 changes © into
If © be of the fourth degree in 6 and k be one of its roots, a transformation
1 1 $2
= k + ^, ©^ = X2-'
leads to an expression of the same kind, where is of the third degree and
so can be taken (after the above) as 4i6^ — gocf) — gs- Moreover, both of these
transformations are birational ; and neither of them affects the general
* Coll. Math. Papers, t. viii, p. 181.
556 ELLIPTIC FUNCTIONS AND [248.
character of the expressions for iv and z, which accordingly can be taken in
the form
iv = A+ B^^, z = G + JJ^^,
where
A, B,G, D are rational functions of <^ ; and ^, 2 are rational functions of w
and z. Now take = ^ (a), where a is a new parametric quantity ; then
- g.^)
must give 2wi + 1 values of . Hence, in the most general case, the degrees
of P, S, R are m, and the degrees of Q, T are m — 1. Thus a curve, of order
2??i + 1 and genus 1, is represented by
_ (<^, i)^'^ + ((^, 1)^-1 # _ ((^, 1)"^ + ((/), ly^-^^i
"^ (^TTr ' ^~ ( = 40^ — ^o(/) — ^s; of course, in particular instances, considerable
simplifications may occur.
Ex. 1. The sextic equation
(j/2 + Qxy + .^2)2 = 1 Qxy {xij + lf
is of genus 1 ; express the A^ariables x and y rationally in terms of a parameter and
the appropriate $^. (Cay ley.)
*
Ex. 2. Likewise express in that form the variables of the equations ^
y'^ = {x-ay{x-hf,
y^={x-af{x-hf,
y
Z=S,(w„z,), W = S,{w„z,) J
express Wj and z^ uniquely in terms of w<2, z^_, and a. Also the relations
w^ = E (Z) + WF (Z), z, = G (Z) + WH (Z))
W=Qj(w,z,a), Z=Q^{w, z,a) \
w = R2 (w^ , z-,), z=^R, (wi , ^1) '
560 TRANSFORMATION INTO ITSELF OF [248.
express Wa and z^ uniquely in terms of w^, z^, and a. The relations therefore
express a birational transformation, and they contain an arbitrary constant ;
hence we have the theorem: An equation, of genus unity, is birationally
related to any other equation, of genus unity and the same invariant modulus,
by equations which involve an arbitrary constant.
Also we infer, as an immediate corollary, that any equation of genus
unity admits an infinitude of birational transformations into itself; the
equations of transformation involve an arbitrary parameter algebraically.
Hence, when j> = l, the number p of p. 545 is unity.
We have seen that, in a birational transformation, an integral of the first
kind belonging to one equation is transformed into an integral of the
first kind belonging to the other. When the genus is unity, each equation
possesses 'only a single integral of the first kind ; and denoting it by v
for the equation f{w, z) = 0, and by u for the transformed equation
ty'2 = 4s'^ — g<2.z' — gz, we have (§ 246)
V = au + b,
where a and b are constants. But u is the argument of the doubly-periodic
functions in terms of which w' and z are expressed. Hence v is effectively
that argument ; in other words, the integral of the first kind associated with
an equation of genus unity is effectively the argument of the doubly-periodic
functions in terms of which the variables are expressible.
lu connection with the preceding discussion, reference may be made to Picard* who
uses the results to prove, among other theorems, that when a differential equation
/(w, i(;') = has integrals w which are uniform functions of z, these integrals are either
(i) doubly-periodic functions of z ; or (ii) rational functions of e^^, where ^ is a constant,
that is, are simjaly-periodic functions of z ; or (iii) rational functions of z. (See also, on
this matter, the Note appended supra at the end of the foregoing chapter x.)
Consider also, in this connection, the birational transformations of the
equation f{w, ^) = into itself, say into f{w', z) = 0. After the preceding
result, we must have some relation
V (w', z') = av (w, z) + b,
where a and b are constants. The integral of the first kind connected
with an equation of genus unity possesses two. distinct periods; let them
be denoted by co^, Wo. It is clear that when v (iv, z) increases by a period,
then V {iv, z) also changes by some period : and likewise for v (w, z),
when V (w', z') increases by a period. Hence we have
\cl>i + ficoo = acoi, X'coi + X coo— awo,
from the first of these results, X, fx, X' , jju being integers : and
&)] = a (pcoi -t ao)..), cOo = a (p'coi + a'o).,),
* Traite cVAnahjse, t. iii, (190S), ch. iv.
248.] AN EQUATION OF GENUS UNITY 561
from the second, p, a, p', a being integers. Denoting Xpf — X'fi by A and
pa' — pa by A' the first two equations give*
Aa)i = a (fx'coi — P'dOo), ^Wo — a (~ X'coi + XcWg).
When these are compared with the second two equations, we have
Lb u — XX,,
^ &)j — ^ 0)2 = pcoi + aco2, —r- a>j + ^co.2 = p aii + a 0)2-
Now the ratio o)^ : w^ is not entirely real (§ 231) ; hence
u. — iJ' — X , \ ,
A = P' -^^""^ ^' = P' A^""'
and therefore
AA'=1.
Now A and A' are integers : hence each of them is 1 or is — 1, that is,
A = ± 1.
Also, let O, denote the ratio co.,: (o^, so that we have
fi'n + x' ^^^
fiCl + X
that is,
/xO- -f (\ -fM')n-\' = 0,
a relation which either is an identity or is an equation satisfied by H.
If the relation is an identity, then
/x = 0, X^ — 0, X = p,'.
Since Xpf — X'p. = ± 1, we have X = p = ±1 (or + i, but these values are to be
excluded because X and p! are integers) ; also a =X, a= pf, that is,
a= ±1.
Hence we have
V {iv\ z')= ±v (w, z) + h.
If the relation is an equation, then the value of ft may not be entirely
real ; consequently
(\ - /)2 + 4 V/x < 0,
that is,
{X + p:y < 4A,
so that A, which is either + 1 or — 1, must now be -f 1 ; and the possible
values ofX + p,' are 0, 1, — 1. Also
^ p,'-X+{{X + p.y-4^/\'l^
2p, ' '
* It is assumed that A is not zero. If A were zero, so that \' = kX, /x'^k/j,, where k is real,
we should have (on dividing one equation by the other]
Xui + fj-ca-i wi'
which would make the ratio of the periods real ; this (§ 231) is impossible.
F. P. 36
562 EQUATIONS OF [248.
which is a non-parametric constant ; so that this case cannot occur if the
invariant modulus of the equation (which is a transcendental function of II)
is parametric. Even in those cases where the invariant modulus has an
appropriate constant value, we have, on eliminating Wj and Wo between
\o)i + /ji(02 = cc(Oi , \'aii + /x'a).2= C('(i>2,
an equation
(k — a) {[x — a) — X'yLt = 0,
that is,
a^ -a{X + fx)+l= 0.
The possible values of X + /u,' being 0, 1, —1, the corresponding values of
a are given by
a^ + 1 = 0, a^ - a + 1 = 0, a^ + a + 1 = 0,
respectively.
In every instance, the constant a is determinate : and all the conditions
are satisfied when the constant h is left arbitrary. Moreover the relations
have arisen in connection with the birational transformation of the curve
into itself; and we therefore infer the theorem that the birational trans-
formations of a curve of genus unity into itself can be represented by
V (w', z')= ±v (w, z) + h,
where h is an arbitrary constant : they obviously constitute the simple
infinitude (p. 560) of birational transformations. The cases, which corre-
spond to a^ -I- 1 = 0, a^ — a -1- 1 = 0, a^ -F a + 1 = 0, are each of them extremely
special.
249. As regards equations (or curves) of genus two, the method adopted
for equations of genus zero or unity can be applied, if we begin with adjoint
polynomials of order n — 3 instead of with those of order n — 2.
It is known that, for equations of genus two, there are two distinct
integrals of the first kind : and each of them determines an adjoint polynomial
of order n — 3. Let these be denoted by U-^ {w, z) and U^ {w, z) ; then as each
of them vanishes to multiplicity X, — 1 at a multiple point of / which is of
multiplicity A,, the polynomial
^ • u. + eu,
also vanishes to that multiplicity, so that, among the intersections ofy"=0
and U^ + 6U2 = 0, such a multiple point counts for \(X — 1) intersections.
Hence the number of intersections of/= and Ui + 6LL= 0, other than the
multiple points of/, is
= n(n-S)-tX{X-l)
= 7i{n-:^)-{(n-l)(n-2)-4>]
in the present case, that is, the number is 2. Eliminate z between the two
equations, and remove from the resulting equation in w the factors which
249.] GENUS TWO 563
correspond to the multiple points off= ; the result is a quadratic in w, the
coefficients of which are rational functions of 0, and the roots of which are
the values of w for the remaining two points of intersection. They are
tv = A + B(&K
where A, B, @ are rational functions, @ having no repeated factor. Proceed-
ing similarly, we find
z = C + D@'K
where C, D, ©' are rational functions ; and substituting in the equation /= 0,
we have ©^ = ©'- , or the variables can be represented in the form
w = A+B@i, 2=C+DbK
where A, B, C, D, S are rational functions of the parameter 0. Moreover
„ f/i (w, z)
©2 =
U.2 (.w, z) '
w- A
the first of these expresses ^ as a rational function of w and z ; the second, on
the substitution of this expression for 6, expresses also ©^ ^g ^ rational
function of tu and z. There is therefore a birational transformation between
the curves
f(w, z) = 0, w'2 = (/) ;
and the curves are therefore of the same genus, viz. 2, the genus of/. Hence
%{z') is of degree either five or six (Ex. 2, § 1'78); and therefore @, as a
polynomial in 6, is of degree either five or six. It therefore appears that
the variables in an equation of genus 2 are expressible as rational functions
of 6 and of the radical ©-, ivhere © is a polynoinial in 6 of the fifth or
the sixth degree.
When is a sextic which (as has been seen) has no repeated factor, it is
of the form
{e-a){e-b){9-c){e-d){e-e){e-f).
Take
(a — b){a — c)
'-a =
©2 =
a — c + (6 — c) is a constant multiple
of ^(1 — 0)(1 — K(^) (1 — X(jb) (1 — fx^) and, in the circumstances, can be
taken as equal to this quantity.
36—2
564 EQUATIONS OF GENUS TWO [249.
When @ is a quintic which (as has been seen) has no repeated factor,
it is of the form
(d -a)(0- h) {6 -c)(e- d) (0 - e).
Take
— a = — (a — h)(p,
which determines a birational transformation : then, as in the preceding case,
^ can be taken as equal to
cj,(l-cl>){l-Kcf>)(l-Xcj>)il-fi4>).
The representation of the coordinates in either case becomes
w = E+F(^i, z^G + H^^,
where E, F, G, H are rational functions of j>, and
= ,/> (1 - c^) (1 - «(^) (1 - X(^) (1 - /X(^).
To determine the degrees of these rational functions, let T be the least
common multiple of their denominators (if any), so that, say,
The line
aw -f /S^' + 7 =
must cut the curve in a number of points equal to its order, and therefore
the equation
{aF + /3R + ryTf = {aQ + /SSy ^
must determine the same number of values of .
If the equation (or curve) / = be of odd order 2m + 1, then P, R, T may
be of degree m, and Q, 8 may be of degree m — 2 ; so that the representation
is
_ ((^, 1)"^ + {4>, l)'"-^ O* ^ (0, l)"^ + ((^, \)rr>-^^h
If the equation (or curve) be of even order 2in, then P, R, 2'' may be of
degree m, and Q, S may be of degree m — 3 ; so that the representation is
_ (ch, ly' + jcf), l)"^-''^ ^ ((f>, iy" + (4>, l)'>^-='#
'^~ {,ir ' ' (<^>i)"
Note 1. The three constants «:, A,, yu, in
determine the three class-moduli, which (p. 545) every equation of genus 2
conserves as invariants under birational transformation.
Note 2. It was proved (Ex. 1, § 246) that the equations
249.] HYPERELLIPTIC CURVES 565
where u^, Ui are a quadratic and a quartic, are birationally transformable into
one another, when they have their class-moduli the same. Similarly, the
equations
are birationally transformable into one another with similar limitations :
where, in each case, neither 11.2 nor Ui has a repeated factor. Now
is geometrically interpretable as a quartic curve : owing to the forms of u^
and u^, the curve has only one double point*, as ought to be the case,
because its genus is 2. Hence any plane curve of genus 2 is birationally
transformable into a plane quartic having only one double point.
Note 3. It might be imagined that a similar result would hold for jj = 3
and for p > 3 : but this is not the case.
In the first place, the argument would not apply. It is true that there
are p adjoint polynomials of order ?i — 3, so that
would be the general equation of a curve of order n — 3, vanishing to the
proper order at the multiple points of /. But the remaining number of
points of intersection is 2p — 2 ; and we should then (if the earlier process
be adopted) have an equation of degree '2p — 2 to solve, its coefficients being
rational functions of p — 1 parameters.
In the second place, if a curve of genus 3 be birationally related to
lU- = / (1 - Z) (1 - K^Z') (1 - K^Z') (1 - K,Z') (1 - K,Z') (1 - K,z),
it manifestly can have only 5 invariant moduli, to determine the five
quantities k. As a curve of genus 3 in general has 6 (=3.3 — 3) such
moduli, it follows that the preceding curve is not general. This argument
applies, a fortiori, when the genus of a curve is greater than 3.
There are curves of genus p, which are birationally related to
w- = / (1 — /) (1 — K^z) ... (1 — K2p--i,z') ;
but they are not general, for they have only 2p — 1 moduli instead of
3j9 — 3. Such curves are often called hyperelliptic.
It thus appears that, so far as concerns the representation of the variables
of an equation in a form that is birational, there is a fundamental distinction
between the cases p < 3, p^ 3. It is also found (though this is beyond the
range of these present investigations) that there is a fundamental distinction
between the properties of functions associated with an equation of genus less
than three and those associated with an equation of genus equal to, or greater
than, three.
* The double point is at infinity.
566 BIRATIONAL TRANSFORMATION OF EQUATIONS [250.
250. We have seen that, in the case of an equation of genus zero, there
is a triple infinitude of birational transformations of the equation into itself
(§ 247); and that, in the case of an equation of genus unity, there is a single
infinitude of similar transformations (§ 248): the infinitude, in each case,
arising through the existence of arbitrary constants in the relations of
transformation. For equations of genus greater than unity, we have the
theorem that the 7iumher of birational trarisforniations of an equation of genus
greater than unity into itself is limited. Schwarz* first proved that such a
birational transformation cannot exist involving an arbitrary parameter, so
that there cannot be a continuous infinitude of such transformations ; Klein -j-
first stated that there could not be a discrete infinitude of such transforma-
tions, that is, not an infinitude of particular transformations ; and Hurwitz]:
obtained 84 (2> - 1) as the upper limit of the number. The first two of these
results can be established by the following argument, due to Picard§ : for the
third, reference can be made to Hurwitz's memoir.
It was proved that, when a birational transformation is effected upon an
equation of genus p, so that it gives another equation also of genus p, the
integrals of the first order associated with one equation are linearly express-
ible in terms of those associated with the other. Let u-^^, ..., Up denote the
normal elementary integrals of the first order associated with f{w, z) = Q;
and let u^, ...,Up' denote those associated with f{w'.z')-=0, a birational
transformation of / into itself. Then we have p equations of the form
where the quantities h and h are constants (§ 246).
The constants h depend only upon the periods. For let the point w, z
move on the Riemann surface from one edge of the cross-cut »« to the same
point on the opposite edge : then the point w' , z' describes a closed irreducible
cycle on its surface. Let HV.s denote the period for w/, which is a combina-
tion of the periods of the normal integrals with integers for coefficients : we
therefore (| 235) have
for all values r, s == 1, 2, . . . , |).
Let U:i_(w, z), U^iw, z), ..., Up{w, z) denote the adjoint polynomials of
order n — 3 arising through the normal integrals : then differentiating the
above relation, we have
duy = kridui + kroduo + ... + krpdup ,
* Grelle, t. Ixxxvii, (1879), pp. 139—145.
t In a letter (dated 1882) to Poincare, quoted Acta Math., t. vii, (1885), p. 10.
+ Math. Ann., t. xli, (1893), p. 424; see also a memoir by him, Math. Ann., t. xxxii, (1888),
pp. 290—308.
§ Cows d'Analyse, t. ii, p. 480.
250.] OF GENUS GREATER THAN UNITY 567
that IS,
Ur (tv', z') -^ = \kn U-, (w, z) + k,.., U^ (w, z)+ ...+krpUp (w, z)] ^. .
dw' dtv
This holds for all the values 1, 2, ..., p, and (by hypothesis) j? > 1. Taking
the relation for r=l, 2, and dividing, we find
C/g (w', z') _ k^i f/i (w, z) + k22 U<2{w, z)+ ... + kip Up (w, z)
Uj (w', z') kn Ui (w, z) + k-^o, Uoiw, z)+ ... + k^p Up (w, z) '
which, in connection with
f{w\z) = 0, f\w,z) = 0,
serves to define the birational transformation of / into itself.
As "the constants k depend only upon the moduli at the cross-cuts of the
Rieraann's surface, so that they are pure constants and are not parametric, it
follows that no arbitrary constant can occur in the equations of the birational
transformation. This is Schwarz's result.
Any birational transformation of /'= into itself leads to a relation (or to
several relations) between the adjoint polynomials of the foregoing type ; and
such a relation may be regarded as the initial form of the birational trans-
formation. In order that the relations may exist, the constants k must
satisfy equations which clearly ax-e algebraical in form. If these equations
do not determine the constants k, then one or more of them would be
arbitrary ; and then the birational transformation would involve an arbitrary
constant, contrary to Schwarz's result. The constants k must then be
determinate, manifestly by a finite number of algebraical equations. Hence
there is a limited number of solutions ; accordingly, as each solution deter-
mines a birational transformation, there is only a limited number of birational
transformations, distinct from one another. This is Klein's result.
The preceding argument is valid, only if p ^ 2. The results are known
not to hold when p <2.
For various properties (such as the periodicity for repeated application) of the birational
transformations of an equation of genus greater than unity into itself, see Hurwitz's
memoirs quoted on p. 566; also Baker's Abelian Fimctions, ch. xxi.
251. The assignment by Riemann (§ 247) of a class of equations, as
constituted by those of the same genus birationally transformable into
one another, suggests the desirability of reserving some form of equation
of that genus as a normal form (or normal curve). When p = 0, a normal
form is superfluous; when jo = 1, the normal form can clearly be taken to be
the nodeless cubic (§ 249) ; when p = '2, the normal form can clearly be taken
to be the uninodal quartic (§ 250) ; and we therefore are concerned with the
cases, where p is equal to 3 or is greater than 3,
568 NORMAL FORM OF EQUATIONS [251.
Denoting the p functions, which arise as the derivatives of the p normal
elementary integrals of the first kind, by c^j, ..., either z' — a or tu' — a begins
with a power of ^ equal to the degree of the cycle and neither of them
begins with a lower power. Accordingly, we can take linear combinations
of X, Y, Z above, so as to secure that p, a, r are unequal, without affecting
the degree of the cycle.
X Y
When the cycle has a finite origin as above, viz. z — a = ^ , tu — a = -^ ,
Z Z
then p > T, (T> T\ the degree of the cycle is the smaller of the two integers
p — r, a — T. The reason for the introduction of homogeneous coordinates is
to treat cycles by one and the same method, whether their origin be in the
finite part of the plane or at infinity. Accordingly, we assume that the three
integers p, cr, r are different from one another and are in decreasing order :
then o- — T is the degree of the cycle. The number p — o- is the class of
the cycle.
When the origin of a cycle is at infinity, the expression of the branches in the cycle is
of the form
■i/=a(,Xp, the three indices in decreasing order are q, q—p, 0: so that q—p is the degree of
the cycle. If q—p, the degree depends upon the lowest index in Y—a^^X, being equal
to that index if less than q. If g- < p, the decreasing order of the indices is q, 0, q —p :
so that p — q is, the degree of the cycle.
572 RESOLUTION OF [252.
For the second case, we have
where l~^\\ as before, we take
and therefore
F-yo^^yiC^ + H....
The indices, in decreasing order, are 2" + ^, g, 0: the degree of the cycle is q.
For the third case, we have
_k fc + l
where ^' ^ 1 ; we take
z-1, z=cs r=c«(i3oC'+...)=/^oC'^''+---
The degree of the cycle is q.
Lastly, y may have infinite values for finite values of x : the expression of the cycle
then is
,« s— 1
y = \x 9 + Si^ 2 +...,
where s > 0. We take
the degree of the cycle is s.
For purposes of transformation, we use the birational transformation
which arises out of a geometrical relation used for this purpose by Halphen*.
Given a plane curve G ; let an arbitrary conic % be taken in its plane :
draw the tangent to G at any point p of the curve, and let this tangent
be intersected at P by the polar of ;p with regard to S ; then P is regarded
as the geometrical transformation of.j?. Manifestly, a point p leads to
a single point P ; to infer the converse, let PP' be the polar of p with
regard to the conic, P' denoting the point where the line touches the
reciprocal of G. Thus pPP' is a self-conjugate triangle ; ^and therefore pP'
is the polar of P. Hence given the locus of P, we find p by drawing
the polar of P with regard to the conic : this will cut (7 in a number of
points: we select as p* that one of the points such that Pp is a tangent
to G at the point f. Thus each point of the locus of p determines a
single point P, and conversely ; the analytical expressions of the geometrical
relation constitute a birational transformation, which may be called Halphen s
transformation.
Two forms arise, according as the conic does not or does cut in real points
the curve to be transformed. The conic is at our disposal and it could be
* Liouville, 3™<= Ser., t. ii, (1876), pp. 87—144.
+ A limited number of pairs of points can exist, for any conic 22, such that the construction
would not discriminate between them. We do not regard this as interfering with the general
character of the transformation : its significance in the result appears towards the close of the
investigation. For the analytical relations, expressing jj in terms of P, see a note by the author,
Messenger of Mathematics, vol. xxx, (May, 1900), pp. 1 — 7; they are not actually required for the
succeeding investigation.
252.] ■ A CYCLE 573
chosen so that it does not cut the curve to be transformed ; but the
transformation has to be repeated a number of times, and some of the
transformed curves might be cut by the conic. On the other hand, there is
a finite limit to the number of times the transformation is applied ; and we
may therefore assume that, if the conic cut in real points the curve or a
transformed curve, the tangents to the conic and the curve are different from
one another.
When the conic and the curve do not cut, take any point M on the curve
and the tangent MM" to C; let M"M' be the polar of M with regard to t,
cutting MM" in M" ; and let MM' be the polar of M" with regard to 2,
cutting M'M" in M'. The triangle MM'M" is self-conjugate with regard
to 2 ; when this is chosen as the triangle of reference, the equation of
the conic can be taken to be
When the conic and the curve cut, say in a point M, then let the
tangent at M to the curve cut the conic in M and M' ; and draw M"M' ,
M"M the tangents to the conic at these points. We choose MM'M" as
the triangle of reference ; the equation of the conic can be taken to be
Y' + 2ZZ = 0.
In the former case, let x, y, z denote the position p, and X, T, Z that of
P. Then we have
Xx+Yy + Zz = 0,
because P lies on the polar of p with regard to the conic ; and
X, Y, Z\ = 0,
X, y,
x, y', :
(where x = dxjdt, and so for y and z), because P lies on the tangent at 'p to
the original curve. Hence
X = y {xy - yx') — z {zx — xz') = x {xx + yy + zz') — x' {jx? + y^ + z")"
Y — z {yz — zy') — x {xy —yx') — y {xx + yy' + zz') — y' {x^ + y' + z^)
Z = X {zx — xz) — y {yz' — zy') — z {xx + yy' + zz) — z {x"^ + y"" + z'^).
which express X, F, Z in terms of x, y, z.
In the latter case, we find similarly the relations
X = y {xy — yx) — x {zx — xz')\
Y=-x {yz - zy') - z {xy - yx) \
Z = z {zx — xz') — y{yz' — zy')j
which • express X, F, Z in terms of x, y, z. These respective relations
constitute part of the analytical form of Halphen's transformation, which is
birational for the curve but not birational for the plane.
574 halphen's [252.
Let any point on the curve to be transformed be denoted by
^=r2^(D=r(To+...)i
where R, S, T denote regular functions of ^ in the vicinity of ^ = : and we
assume that the integers p, a, r are distinct from one another.
When the second transformation is the one to be effected, we remember
that the tangent to the curve is the axis of y for the triangle of reference ;
accordingly, we assume o- > /o > t, so that p — t is the degree of the cycle, and
cr — p is the class of the cycle ; also p — 2o- + t < 0. The result of the
transformation is *.
X = - Po^To (p - t) r^P+^l + . . . = p/'^P" + . . . >
Y = poCToT, {p-2a + t) ^p+-+-i + . . . = a-o"r" + . ,
Z = To^po (P - t) ^''+'^-' +...= To"r" + . .
say. We have a" > p" > r" ; also p" — t" = p — r, a" — p" = a — p ; so that the
degree and the class of the cycle (if any) are unaltered.
When the transformation with regard to an assumed conic is applied
a number of times, the conic being assumed so as not to cut the initial curve,
then it may happen that, after a number of transformations, the latest curve
cuts the conic. The further application of the transformation will then, by
the foregoing analysis, have no effect upon either the degree or the class of a
cycle : it therefore is ineffective for our purpose of further reduction. To
secure such further reduction, we should proceed to effect transformation
with regard to another arbitrarily assumed conic, chosen so as not to cut the
curve.
When the first transformation is the one to be effected, assume that
p > o- > T, so that o- — T is the degree of the cycle (if any) at the point, and
p — o- is its class. The result of the transformation is
x = ^<'^^—^{p,T,^{T-p) + ...] = i:^'{p:+...y
Y= ^<^+2-i |^^^^2(^ -a)+...]=^^' {a: + . . .)
Z= ^2^+^-1 {croVo(o- - t) +...} = r (To' + ...))
say. We have p > a' \ also because t' — a = a — t, we have t > a' ; there
are therefore three cases.
(i) Let p' >t' > cr'. The degree of the cycle (if any) is r' — cr', that is,
o" — T : it is unaltered by the transformation. The class of the cycle is p' — t',
which is p — 2(7 + T, = p - cr — (cr — r), so that p —t' < p — a; it is decreased
by the transformation. In this case p — 2o- + t > ; also
p' — 2t'+c7', =p — 2cr-t- t — (cr — t), p > a'. The degree of the cycle (if any) is p — a , that is,
p — (T. But t' > p', so that p — a< (t — t: hence p' — a < a — r, or the degree
is decreased. The class of the cycle is t — p', that is, a — t — (p — a); there
is no useful rule of increase or decrease compared with the old cycle in
general, though we note that the new class is less than the old degree. In
this case p — 2cr + t < 0.
(iii) Let t' = p', so that p — 2cr + t = 0, or, the degree and the class of
the cycle (if any) are equal. Effect a (birational) transformation
X' = X, Y' = Y, Z' == aif' (a — t) X — PqTo {'^ — p) ^
= r' (To'" +•..).
where r'" > p + 2t—1, that is, r'" > p . Accordingly, for the transformed cycle,
we have r"' > p > a-'. The degree of the cycle (if any) is p — a' , that is, p — a,
which is equal to cr — t : it is unaltered by the double transformation. Also
t'" - 2p' + p + 2t - 1 - 2 (p + 2t - 1) + a- + 2t -1 > a - p,
where o- — p is a negative quantity. If t" — 2p' + o-' > 0, another application
of the transformation by the conic leads to the first case above : and the
class is decreased, while the degree is unaltered. If r" — 2p' + cr' < 0, another
application of the transformation by the conic leads to the second case above :
the degree is decreased. If r'" — 2p + cr' = 0, the new class is equal to the
new degree, each of them equal to the common value of the old class and the
old degree : the cycle must be considered further.
For this last case, let p — cr = cr — r = n : then
f-f-(^■■■)■ f=^"(^••■)■
or, remembering the source of the homogeneous coordinates, and taking
^ : q : \ = y : X : z.
we have
^ = pG^(r)>
where (r is a regular function of |", that does not vanish with ^ : and this
corresponds to the original cycle by a birational transformation. It may
happen that the initial powers of p' in G give integral powers of ^ ; let
^ = P(l)+f^'^G^i(r),
where P (^) contains powers of ^ not higher than the gth and not lower than
the second, where 7^ > a > 0, and where G^ does not vanish with ^. To this
form, apply the Halphen transformation with respect to the conic
, G {x, y, z) = 0,
576 RESOLUTION OF A CYCLE [252.
taken arbitrarily in the first instance. The point X, Y which corresponds to
f , 7] is given by
F-, = ^(X-f)=,-(X-|);
OX oy oz
where we write x, y, z = r), ^, 1 in the latter : thus
X=a(^,7],v'), Y=^(^,v,v\
where a and /3 are rational functions in r]' of the first degree. Hence
dY , ^^ + /3^V + /3vV' ,. , ..
d'Y
dX'"
generally
dX
dy
d|
dX
d^
— 7 '
\i^ '/) '/ 5
dy
dt]"
'/
and
«f +
a,^' + S
87
d'^Y ?r)"
Choose the conic C, so that
^, =1= 0, a^ + a^i] + a,/ r?" ^ 0,
for ^=0, 17 = 0; then as 7;<9'+^> is the first derivative of 77 with regard to ^
that becomes infinite at ^ = 0, 7; = 0, so F'^' is the first derivative of Y with
regard to X that becomes infinite at the new origin : that is, if Fq, X^ be the
new origin, we have
Y-Y, = A (X - Xo) + (X - Xo)'"'^- G, {(X - Xo)-}.
Let the transformation be applied g- — 2 times in succession ; we have, at the
end,
F' - Fo' = a, {X' - Xo') + a, (X' - Xo')^ + a, (X' - Xo'f^" + . • • ,
or changing the axes by taking X' — Xq = X", Y' — Y^ — a^ (X' — Xq) = Y",
we have
F" = a,X"^ + a3X"'^'"^+...,
where n > a> 0. This can be represented in the homogeneous form
y = ^'\ X = a^^-" + a3^2n+a ^ _ _ _ ^ ^ ^ 1 = ^o_
Applying now the Halphen transformation in its analytical form, we have
X = - 2na2^-»-^ - (2?2 + a) a^l;-''+^-^ - ...,
Y^-n^""-^- ...,
Z = n ^-^-^ + 2nai ^'''-' + ...;
252.] INTO LINEAR CYCLES 577
or transforming so as to have X + la^Z, Y, Z, as the new coordinates, we
have
X'" =-{2n + a) ttst^^+^-i - . . .1
The three indices in decreasing order are 2u + a— 1, 2« — 1, 7i — 1 ; the
degree of the cycle is n, its class is a, which is less than n ; and therefore the
birational transformations have reduced the class below the degree.
Hence given a cycle of any degree m, greater than unity, and of any class
7)i.', we can by Halphen's birational transformation change it into another
cycle. If m be greater than in, the new class is less than m while the new
degree is m : and repetition of the transformation can, by the first case, be
made, so long as the class is greater than the degree : and, by the third case,
until the class is less than the degree ; without altering the degree. When
another repetition of the transformation is made, the degree will (by the
second case) be decreased.
Proceeding in this way, we can make the cycle of the first degree : then
by the first case, of the first class also : that is, we can make the cycle linear.
When this process is applied to each cycle, the final equation has only linear
cycles : and it is connected with the initial equation by birational transformation.
Further, it was proved that a Halphen transformation of a linear cycle of
the form
w — a == A'l {z — a)-\- ko {z — a)'' + • • . + ^m (z — a)'" + . . .
leads to a relation
W-a'= k,' (Z - a) + h' (Z-a'y+...+ k'„,_, (Z- a')^-^ +...,
where /c,,„,_i depends upon ^■,^ linearly and upon k^^, ..., A'-m^j. If therefore
two linear cycles agree up to (but not beyond) the nith order of small
quantities, the transformation replaces them by two linear cycles agreeing up
to only the (???, — l)th order of small quantities: and so on, by successive
repetitions of the transformation, until they agree only in their origin, so that
the tangents differ. Now the origin of a new cycle, engendered by trans-
formation, lies on the tangent to the cycle which is to be transformed : hence,
applying a Halphen transformation once more, the origins of the two cycles
are different. It therefore follows that the cycles of any degree and class
having a common origin can be birationally transformed into linear cycles,
each of them with its own origin distinct from that of all the remainder.
The resolution thus effected transforms every multiple point 'of any
charaeter into an aggregate of simple points, and would therefore transform
an equation with multiple characteristics into an equation having only simple
points, if new singularities were not introduced in the process of birational
transformation. Reverting to the initial geometrical exposition of the
birational transformation, we see that such singularities may arise through
P. F. 37
578 BIEATIONAL TRANSFORMATION [252.
exceptional points on the locus, as follows. From P, which is uniquely
obtained from the point p on the given curve, a number of other tangents
can be drawn to that curve ; let q denote the point of contact of any one of
them. In order that pq may be the polar of P with regard to the conic, one
relation must be satisfied by x and y, the coordinates of P, in addition to
the equation of the curve on which p lies : that is, there are two algebraic
equations satisfied by {x, y), and therefore there is a finite number of
simultaneous values satisfying these conditions. Each such point is a double
point on the locus of P, with distinct tangents for the branches : so that the
transformed curve thus possesses simple nodes unrepresented by any multi-
plicity on the original curve. But, in general, there cannot be two points
such as q, say q and q, the tangents at which are concurrent with the tangent
at p ; for this purpose, some relation between the coefficients of the curve and
the constants of the conic would need to be satisfied — which is not the case,
when the conic is arbitrarily assumed and the curve is not extremely special.
We therefore have the theorem, expressed geometrically :
A7iy algebraic curve can he hirationally transformed into some algebraic
carve the singularities of luhich are double points with distinct tangents.
The two curves must be of the same genus, and they must have the same
moduli : the complexity of the transformation manifestly depends upon the
original equation.
Ex. Consider the resolution* of the singularity of
if - "ix^f + x^ = %x°y^ '
at y = 0, « = 0.
Proceeding as in Chap. VIII., we find the six branches of the curve given by
where co^--!, &^ = \ : thus for our homogeneous coordinates, we may take
Thvis p = 8, (r=6, r = 0. The class of the cycle is 2, the degree is 6.
Since p — 2o- + r<0, Halphen's transformation, when applied, falls under the second
case. The indices after transformation are
t' =2(7+ r-l = ll = pi']
p' = p + 2t - 1 = 7 = (Ti> ;
0-'= o- + 2r-l= 5 = rJ
the class of the cycle is 4, the degree is 2.
Since pj — 2o-i+ri >0, Halphen's transformation, when applied, falls under the first
•case. The indices after transformation are
p"= pi + 2ri-l = 20j
r"=2cri+ ri-l = 18
0-"= o-i + 2ri-l = 16j
* C. A. Scott, A7ner. Journ. Math., t. xiv, (1892), p. 318.
252.] OF ALGEBRAIC CURVES 579
the degree of the cycle is 2, the class is 2. For the cycle, we have
and therefore
Y" = A^X"^ + ....
This is an instance of the second case, when the class and the degree are equal : the
actual form is
4 81 47^43 ,„ V-,
9 4 8
Three applications of the Halphen transformation will reduce the class to unity and will
give a cycle of the form
,, , -. ,1^
y =a2x''-' + a^x "^+...;
and one more application will give a linear cycle.
I^ote. The preceding sketch (§§ 245 — 252) is intended only as an intro-
duction to the theory of birational transformation, the development of which
really belongs to the detailed theory of Abelian functions.
Moreover, transformations which are rational but not birational are
practically omitted from consideration. If further information be ) W (iV, Z) ,
^ 4dz - A
M{z)
¥
= ©
M{z)C{z)j
C(z) 2 — ^^7 — •' log 6 (wr, z) I ,
dWr
where ^ is a constant of integration, and where we now write w for w-^
in the subject of integration, Wi having been any branch of the quantity w
defined by f{iu, z) = 0. To evaluate the right-hand side, only algebraic
expansions are necessary; and the result will be some function of the
parameters. These parameters are connected by the equation P (x) =
* The symbol, in this significance, is due to Boole, Phil. Trans. (1857), p. 751 ; the various
coefficients are manifestly the Cauchy residues (§ 25, Ex. 9) of the expression for its poles, which
arise through the zeros of ill (z) C (z) and through an infinite value of z.
586 Abel's theorem [252.
with the upper limit of the integrals : when expressed in terms of them, the
right-hand side is clearly a logarithmic and an algebraic function of those
limits, which in special cases may degenerate to a more simple form. The
constant A can, if desired, be determined by taking another conditional
equation, similar to (lu, z) = 0, in order to assign lower limits to the
integral.
This result is Abel's Theorem in its most general form. We proceed
to some applications, first recapitulating the significance of the various
symbols. The fundamental equation is f{w, z) = 0, of degree n in w,
having the coefficient of w''^ equal to unity, and polynomials in z for the
coefficients of the remaining powers of iv. The conditional equation is
6 (w, z) = 0, which (by means of /= 0) is taken of degree not higher than
7?. — 1 in iu\ the coefficients of the various powers of %u are polynomials
in z, having arbitrary parameters for coefficients of powers of z. The
result of eliminating w between /=0, ^ = 0, is obtained; G {z) represents
the aggregate of factors, corresponding to roots that are independent of the
parameters; the roots, that depend upon the parameters, are taken, in
conjunction with the appropriate values of iv, to be the upper limits of
the integral. The quantity W {iv, z) is a polynomial in iv and in z, of degree
not higher than ?i — 2 in iv. The quantity M {z) is a polynomial in z\
it may be a constant ; if it is variable, no one of its roots may be one of the
quantities x^, ..., x^ belonging to the upper limits of the integrals. And
lastly, the symbol @ requires the various algebraic operations specified
in its definition, connected with the roots of if (^) = and of G {z) =
as well as with an expansion in descending powers of z.
Usually we have G {z) = l; but even when it is different from unity,
its roots frequently contribute only zero terms to the final sum on the right-
hand side.
Note. The preceding proof dispenses with many of the properties of functions of
position on a Kiemann's surface that have already been estabhslied ; the main reason
why such a proof is given is, that some notion of Abel's theorem may be obtained on the
lines solely of Abel's analysis. We shall, however, in the proof of other results, use more
freely the properties of functions of position to which reference has just been made.
Another method* of obtaining the result is to consider the integral
taken over the Riemann's surface, where / denotes
' "-N g W {VJ, Z) ^_
M(z^S
ow
If is left as an exei'cise : it follows the lines of § § 230 — 238.
* This is Neumann's method, Vorlesungen ilber Eiemann's Theorie der AbeVsehen Integrale,
pp. 285—303.
252.] EXAMPLES 587
Ex. 1. Let the permanent equation be
and take
aw + hz-\-c=0
as the conditional equation. There are three quantities .rj, a'2, x^ given as the roots of
a?{Az^-g^z-g^)-{hz + cf = 0.
Now take the integral of the first kind associated with the equation ; it is
^dz
"We have M{z) = \, C{z) = l, W{;w, z) = \; and the roots of the permanent equation can be
taken as w\ —vf. Hence in the operation we have only to obtain the coefficient of
- in a descending expansion, so that
z
s /"'^o-fis d 1 \
2 / A=-Cx\-\og{aw->rhz-\-G) \Qg{-aw-{-hz-\-c)\
= lj ''<' -.. \y^ w J
o-=l
~ 1 , hz-\-c-\-aw
= - (7i - log -. .
r 10 oz + c — aw
When the quantity on the right-hand side is expanded in descending powers of z, the first
ttZ 1
term is — - , so that no term in - exists. Accordingly
2z^ ^
3 /■ '"-V dz
where ^ is a constant independent of the arbitrary constants in the conditional equation,
and therefore determinable by assigning special values to those constants. Taking a=0,
6 = 0, the three values oi x become each infinite, so that
3 /"-Vri^ ^
2 - = 0.
Now let x^ = p {u^), for a= 1, 2, 3 ; the last equation can be written
i
rw,
P{a),
P'M ,
p (^'1),
F(«i),
P{U2),
F(«2),
where x^, x0, being a positive integer unless z= qo is a branch-point.
Hence, in the vicinity of ^^ = oo , we have
W (w, z) _ - kA-^
Moreover, log Q (w, z), when expanded in descending powers of z, contains no
term zf having a positive integer for its index p. Consequently, in
W{iu, z) ^ ar \
(7i — ^^ log e (w, z)
no term in - occurs, for any of the branches w ; and therefore
. ^ of
Hence
^M W (w, z)
dz = K,
dw
where K is a constant of integration independent of the parameters in
e (w, z).
The constant K can be obtained by taking the sum of the integrals
as determined by a different set of parameters in 6 which (as ^ is a
constant independent of their value) may be made as particular as we
please. Moreover, we recall the property that an integral of the first
kind upon a Riemann's surface is not entirely determinate, its expression
being subject to additive integral multiples of its periods: and we have
the theorem that the sum of the values of an integral of the first kind
at the positions on a Riemann's surface, tvhere a polynovnal 6 (w, z)
vanishes, remains unaltered no matter hoiu the parameters in that poly-
nomial are changed: subject always to modification by additive multiples
of the periods. The theorem is also expressed sometimes as follows : If
u (w, z) denote an integral of the first kind, then
1 v{w^,z„.)= S u{y I ' . dz = S
At
(z - of
,
rational on the Riemann surface: then
^ E{w„,z,)- ^ E{ya,x^) = -A -^^ — ^
The equality is subject to additive multiples of the periods of the integral.
Ex. 1. lu the preceding investigation, the assumption is made that a is finite, so that
?=ao is a point where the integral is finite.
Obtain the corresponding result when 2 = oc is the sole simple infinity of a normal
elementary integral of the second kind.
252.] THE ELEMENTARY INTEGRALS 597
Ex. 2. The line aw 4-/32 + 7 = touches the curve f{^o, z)=0, and U {w, z) is an
adjoint polynomial of order n — 2, which vanishes at. each of the remaining n — 2 points
•where the line meets the curve. Obtain an expression for
M f'^a- U {W, Z) dz
where the upper limits of the integrals are the points of intersection oif{w, z) = with any
other curve 6 {w,z)=0; also an expression for
U {w, z) dz
are, by the two equations of condition, the same for ds and
dS, and denoting by t/t the real quantity (<^ — 6")^, we have
ds^ = (a' + b^ + C-) {dt + du (6 + ?»} [dt + du (6 - ?»},
and dS-^= (A"" + 5^ + C^) [dt + du(e + ;»} [dt + du (9 - ?»}.
253.] OF SURFACES 605
Then, except as to factors which do not involve infinitesimals, the factors of
cZi'- and of dS^ are the same. Hence, except as to the former factors, the
numerator of the fraction for m^ is, qua function of the infinitesimal
elements, substantially the same as the denominator; and therefore either
., dP + idQ .dP-idQ „ ., .. . . .
(a) , . , and , r^- are finite quantities snnultaneouslv ;
dp + idq dp — idq ^ -^
or
.^,, dP + idQ ,dP-idQ . ., . . . . .
(p) --. r-^ and r^ r^ are finite quantities simultaneously.
dp — idq dp + idq -^
Either of these pairs of conditions ensures the required form of in, and so
ensures the conformal similarity of the surfaces.
Ex. Shew that both p and q satisfy the partial difierential equation
Consider {a) first. Since {dP + idQ)/(dp + idq) is a finite quantity, the
differentials dP + idQ and dp + idq vanish together, and therefore the quan-
tities P 4- iQ and p + iq are constant together. Now P and Q are functions
of 'the variables which enter into the expressions for_p and q; hence P + iQ
and p + iq, in themselves variable quantities, can be constant together only if
P + iQ=fip + iq),
where /denotes some functional form. This equation implies two independent
relations, because the real parts, and the coefficients of the imaginary parts,
on the two sides of the equation must separately be equal to one another ;
and from these two relations we infer that
where / {p — iq) is the function which results from changing i into — i
throughout /(p + iq) and is equal to f(p — iq), if i enter into /only through
its occurrence in p + iq. From this equation, it follows that
dP - idQ
dp - idq
is finite ; and therefore a necessary and sufficient condition for the satisfaction
of (a) is that P, Q and p, q be connected by an equation of the form
P + iQ=f(p+iq)-
Moreover, the function / is arbitrary so far as required by the preceding
analysis; and so the conditions will be satisfied, either if special forms
of / be assumed or if other (not inconsistent) conditions be assigned so
as to determine the form of the function.
Next, consider (/3). We easily see that similar reasoning leads to the
conclusion that the conditions are satisfied, when P, Q and p, q are connected
by an equation of the form
p + iQ = g(p- iq) ;
606 gauss's [253.
and similar inferences as to the use of the undetermined functional form of g
may be drawn. Hence we have the theorem : —
Parts of two surfaces may he made to correspond, point by point, in
such a way that their elements are similar to one another, by assigning
any relation between their parameter's, of either of the for-ms
P + iQ=f{p + iq), P + iQ^g(p-iq);
and every such correspondence between two given surfaces is obtained by the
assignment of the proper functional form in one or other of these equations.
Ex. In establishing this conformal representation, only small quantities of the first
order are taken into account. Sketch a method whereby it would be possible to evaluate,
to a higher order of small quantities, the magnitude
dS' ds'
where dlS, dS' are two small conterminous arcs on one surface, and ds, ds' are the
corresponding small conterminous arcs on the other surface. (Voss.)
254. Suppose now that there is a third surface, any point on which
is determined by parameters \ and /ju ; then it will have conformal similarity'
to the first surface, if there be any functional relation of the form
\ + i/jb = h(p + iq).
But if h~'^ be the inverse of the function h, then we have a relation
P + iQ=f{h-'(X + ifM)]
= F{\ + ifi),
which is the necessary and sufficient condition for the conformal similarity
of the second and the third surfaces.
This similarity to one another of two surfaces, each of which can be made
to correspond to a third surface so as to be conformally similar to it, is an
immediate inference from the geometry. It has an important bearing, in
the following manner. If the third surface be one of simple form, so that its
parameters are easily obtainable, there will be a convenience in making it
correspond to one of the first two surfaces so as to have conformal similarity,
and then in making the second of the given surfaces correspond, in conformal
similarity, to the third surface which has already been made conformally
similar to the first of them.
Now the simplest of all surfaces, from the point of view of parametric
expression of points lying on it, is the plane : the parameters are taken to
be the Cartesian coordinates of the point. Hence, in order to map out two
surfaces so that they may be conformally similar, it is sufficient to map
out a plane in conformal similarity to one of them and then to map out
the other in conformal similarity to the mapped plane : that is to say, we
254.] THEOREM 607
may, without loss of generality, make one of the surfaces a plane, and all
that is then necessary is the determination of a law of conformation.
We therefore take P = X, Q=Y,N=1: and then
P + iQ = X + iY=Z,
where Z is the complex variable of a point in the plane ; and the equations
which establish the conformation of the surface with the plane are
ds' = n (dp- + dq^) •
X + i7=f(p + iq)
m^n =f (p + iq)f' (p - iq)
where f I (p — iq) is the form of f(p + iq) when, in the latter, the sign of i is
changed throughout.
As yet, only the form P + iQ = f(p+ iq) has been taken into account.
It is sufficient for our present purpose, in regard to the alternative form
P + iQ = g{p — iq), to note that, by the introduction of a plane as an inter-
mediate surface, there is no essential divStinction between the cases*. For
as P = X, Q = Y, we have
X-YiY=g {p-iq),
and therefore X — iY= g-i^{p + iq),
which maps out the surface on the plane in a copy, differing from the copy
determined by
X ^iY = g,{p + iq),
only in being a reflexion of that former copy in the axis of X. It is therefore
sufficient to consider only the general relation
X + iY=f{p-^iq).
Ex. We have an immediate proof that the form of relation between two planes, as
considered in § 9, is the most general form possible. For in the case in which the
second surface is a plane, we have ds^ = dx''- + dif, so that n = l, p = x, q=y: hence the
most general law is
X+iY=f{x + iy),
that is, w—f{z),
in the earlier notation. Some illustrations arising out of particular forms of the function
/will be considered later (§ 257).
255. In the case of a surface of revolution, it is convenient to take <^
as the orientation of a meridian through any point, that is, the longitude of
the point, a as the distance along the meridian from the pole, and q as
the perpendicular distance from the axis ; there will then be some relation
between cr and q, equivalent to the equation of the meridian curve. Then
ds"^ — da- + q-d^'^
^ q^ (d(f>' + d0%
* A discussion is given by Gauss, Ges. Werke, t. iv, pp. 211 — 216, of the corresponding result
when neither of the surfaces is plane.
608 ' CONFOEMAL REPRESENTATION OF [255._
where dd ■= — , so that ^ is a function of only one variable, the parameter of
the point regarded as a point on the meridian curve. Here n=q"; and so
the relation, which establishes the law of conformation between the plane
and the surface in the most general form, is
*' + iy =/(<^ + *'^) ;
and the magnification m js given by
m^ q^ =f ((b + ie)f,' (0 - id).
Evidently the lines on the plane, which correspond to meridians of
lono'itude, are given by the elimination of 6, and the lines on the plane,
which correspond to parallels of latitude, are given by the elimination of + ie) -/,{ -ie)\-
Ex. 1. A plane map is made of a surface of revolution so that the meridians and the
parallels of latitude are circles. Shew that, if (r, a) be the polar coordinates of a point on
the map determined by the point {B, (p) on the surface, then
?^^ = - 2ac {ae^"^ cos 2 {c^ +g) + b cos {g + h)},
r
?^ = 2ac {ae^'^ sin 2 {c^ +g) + b sin [g + h)],
where a, 6, c^g, h are constants.
Prove also that the centres of all the meridians lie on one straight line and that the
centres of all the parallels of latitude lie on a perpendicular straight line. (Lagrange.)
Ex. 2. Prove that, in a plane map of a surface of revolution, the curvature of a
meridian at a point <9 is ;^ ( — ) , and the curvature of a parallel of latitude at a point nqj
circles on the plane map given by
z=f{+ie),
the function /and the conjugate function /j must satisfy the relation
{f,^ + ie}=-{fu^- + d'^%
where sech ^ = cos \. Hence we have
X + iY=f((j> + i^);
and the magnification m is given by
ma cos \ = {/' {(j) + ^^)// (q6 - i^)}^. .
There are two forms of / which are of special importance in representa-
tions of spherical surfaces.
First, let/(/Lt) = k/x, where k is a real constant ; then
X + iY=k(cf> + i^),
and therefore X = k(f), Y= k^ = k sech~^ (cos X) ;
that is, the meridians and the parallels of latitude are straight lines,
necessarily perpendicular to each other, because angles are conserved.
The meridians are equidistant from one another ; the distance between
two parallels of latitude, lying on the same side of the equator and
having a given difference of latitude, increases from the equator. We
have /' ((f) + i^) = k =// (<^ — i^) ; and therefore
m = ~ sec A,,
a
or the map is uniformly magnified along a parallel of latitude with a
magnification which increases very rapidly towards the pole. This map is
known as 3Iercator's Projection.
Secondly, let/(yu,) = ke^'''^, where k and c are real constants ; then
' X + iY= ke^" <*+^^' = ke-"'^ (cos c^ + i sin c0),
and therefore X = ke'"^ cos ccj) and F= ke^"'^ sin ccf).
For the magnification, we have
/' ((/) + {^) = icke^'^ '*+'"^' and // ((/> - t^) = - icke-''^ <*-^'^',
so that ma cos X. = cke~'''^,
ck ,, , c^^(l-sinX)4(''-^)
or • m = — e "^ secX = — 7^7— — r , .
F. F. 39
610 MAPS [256.
The most frequent case is that in which c = 1. Then the meridians are
represented by the concurrent straight lines
F=Z tan ;
the parallels of latitude are represented by the concentric circles
1 — sin A,
)f the circ
lines ; and the magnification is
1 + sin X '
the common centre of the circles being the point of concurrence of the
k
a(l + sin A.) "
This map is known as the ste7^eographic projection. The South pole is the
pole of projection.
It is convenient to take the equatorial plane for the plane of z: the
direction which, in that plane, is usually positive for the measurement of
longitude, is negative for ordinary measurement of trigonometrical angles.
If we project on the equatorial plane, we have
which gives a stereographic projection.
Ex. 1. Prove that, if x, y, z be the coordinates of any point on a sphere of radius a and
centre the origin, every plane representation of the sphere is included in the equation
for varying forms of the function /.
Ex. 2. In a stereographic projection of a sphere, the complex variable of a point
corresponding to a point Pon the sphere is x-\-%y. Prove that the complex variable of the
point, which corresponds to the point diametric'ally opposite to P, is .
Ex. 3. Two conforraal representations of the surface of a sphere on a plane are given
by Mercator's projection and a stereographic projection. Find the form of relation which
will transform these projections into one another.
Ex. 4. Shew that rhumb-lines (loxodromes) on a sphere become straight lines in
Mercator's projection and equiangular spirals in a stereographic projection.
Ex. 5. A great circle cuts the meridian of reference (0 = 0) in latitude a at an angle a;
shew that the corresponding curve in the stereographic projection is the circle
{X-\-h tan aj' + ( F+ h cot a sec cCf = k"^ sec^ a cosec^ a.
Ex. 6. A small circle of angular radius r on the sphere has its centre in latitude c and
longitude a ; shew that the corresponding curve in the stereographic projection is the
circle
'' „ k cos c cos a\ 2 / ^^ k cos c sin a\^ k^ sin^ r
^ + T--- + M + 7-^ =/ r-- — N2-
cosr + smc/ \ cos ?• + sine/ (cosr+smc)^
256.] GENERAL REFERENCES 611
The less frequent case is that in which the constant c is allowed to
remain in the function for the purpose of satisfying some useful condition.
One such condition is assigned by making the magnification the same at
the points of highest and of lowest latitude on the map. If these latitudes
be Xi, \2, then
( 1 - sin \j)i "'-^' _ (1 - sin >^)^ '°-i'
(1 + sinXi)*'"^" ~ (1 + sin \,)i^<=+'^ '
so that
1 — sin XA , /l + sin X^
1 fi- — sm XA ,
^°Hr3^mxJ + i°g
1 + sin \.
, , 1 — sm XA 1 / 1 + sin X,
logU— T-;rr -log
J — sm Xg/ \1 + sm Xg/
This representation is used for star-maps : it has the advantage of leaving
the magnification almost symmetrical with respect to the centre of the map.
Ex. Prove that the magnification is a minimum at points in latitude arc sin c.
Shew that, if the map be that of a belt between latitudes 30° and 60°, the magnification
is a minimum in latitude 45° 40' 50"; and find the ratio of the greatest and the least
magnifications.
Note. Of the memoirs which treat of the construction of maps of-
surfaces as a special question, the most important are those of Lagrange*
and Gauss f. Lagrange, after stating the contributions of Lambert and of
Euler, obtains a solution, which can be applied to any surface of revolu-
tion ; and he makes important applications to the sphere and the spheroid.
Gauss discusses the question in a more general manner and solves the
question for the conformal representation of any two surfaces upon each
other, but without giving a single reference to Lagrange's work : the
solution is worked out for some particular problems and it is applied, in
subsequent memoirsj, to geodesy. Other papers which may be consulted
are those of Bonnet§, Jacobi||, Korkinelf, and Von der Miihll**; and there
is also a treatise by Herz"f*f.
But after the appearance of Riemann's dissertation JJ, the question
ceased to have the special application originally assigned to it; it has
gradually become a part of the theory of functions. The general develop-
ment will be discussed in the next chapter, the remainder of the present
* Nouv. Mem. de VAcad. Roy. de Berlin, (1779). There are two memoirs : they occur in his
CGllected works, t. iv, pp. 635 — 692.
t Schumacher's Astr. Abh. (1825) ; Ges. Werke, t. iv, pp. 189—216,
X Gott. Abh., t. ii, (1844), ib., t. iii, (1847); Ges. Werke, t. iv, pp. 259—340.
§ Liouville, t. xvii, (1852), pp. 301—340.
II Crelle, t. lix, (1861), pp. 74—88 ; Ges. Werke, t. ii, pp. 399—416.
Ii Math. Ann., t. xxxv, (1890), pp. 588—604.
** Crelle, t. Ixix, (1868), pp. 264—285.
ft Lehrbuch der Landkartenprojectiunen, (Leipzig, Teubner, 1885).
Xt " Grundlagen fiir eine allgemeine Theorie der Functionen einer veranderlichen complexen
Grosse," Gottingen. 1851 ; Ges. Werke, pp. 3 — 45, especially § 21.
39—2
612 • EXAMPLES [256.
chapter being devoted to some special instances of functional relations
between lu and z and their geometrical representations.
The following examples give the conformal representation of the respective surfaces
upon a plane or a part of a plane.
Ex. 1. A point on an oblate spheroid is determined by its longitude I and its
geographical latitude /x. Shew that the surface will be conformally represented upon a
plane by the equation
for any form of the function /; where sech = cos ju, and e is the eccentricity of the
meridian.
Also shew that, if the function / be taken in the form f{u) — ke^'^, the meridians in
the map are concurrent straight lines, and the parallels of latitude concentric circles ; and
that the magnification is stationary at points in geographical latitude arc sin c. (Gauss.)
Eoc. 2. Let the semi-axes of an ellipsoid be denoted by p, (p^ — b^)"^, (p^-c^)"^, in
descending order of magnitude. Shew that the surface will be conformally represented
upon a plane by the equation
V , -TT V- fz. / , ■\,l^ Q(u + a)e{iv + a)]
-^ \ ^ ^ - ° e{u-a)e{tv-a)l
for any form of the function /; 'where u and v are expressed in terms of the elliptic
coordinates p^ and p2 of a point on the surface by the equations
cHpi'-b^) _ 2 l{pl^)_ 2-
p /c2-62\4
the modulus is - ( —^ — ^r, ) , the constant a is given by
6 = c dn a,
and the value of the constant h is tn a dn a - Z (a). (Jacobi.)
Ex. 3. The circular section of an anchor-ring by a plane through the axis subtends an
angle tt — 2e at the centre of the ring, and the position of any point on such a section is
determined by I, the longitude of the section, and by X, the angle between the radius from
the centre of the section to the point and the line from the centre of the section to the
centre of the ring.
Shew that, by means of the equations
l = 2nx,
tan ^X = cot Je tan (ttt/ tan e),
the surface of the anchor-ring is conformally represented on the area of a rectangle whose
sides are 1 and cot e. (Klein.)
Ex. 4. Consider the surface generated, by revolution round the axis of ?/', of the curve
whose equations are
.^■' = a sin t, i/ = a (cos t + log tan y),
sometimes called the tracti'ix.
The radius of curvature of the curve is — a cot t ; the length of the normal intercepted
between the curve and the axis of y' is a tan t. Hence the Gauss measure of curvature of
the surface of revolution is — 1/a^, that is, the surface of revolution is one of constant
negative curvature. Surfaces of the same measure of curvature can be deformed into one
256.] EXAMPLES 613
anotlier ; in particular, when the measure is constant, the surface is applicable upon itself
in an infinite variety of ways*.
The arc-element of the surface of revolution is given by
= d^ cot^ t dt"^ + or sin^ t dcf)'-^
Let a new variable x//' be introduced by the relation
so that
then the arc-element is given by
When we write
this becomes
, , cos t ,
a\l/- = 7—r- dt,
^ sin t
(/) = .r, >/^ = ?/,
ds^ = —^ {dx^ + d'lf'
and io the surface can be represented conformally upon the x^ y plane.
For the upper half of the surface, corresponding to the positive part of the x' , y' plane
of the original curve, the range of ;; is from tt to ^tt ; and therefore the range of y is from
00 to 1. The range of x is from to Itv. The area in the x, y plane is a half -rectangle ; it
is bounded by a line x=Q> while y ranges from co to 1, by a hne y = \ while x ranges from
to 277, by a line x=^tt while y ranges from 1 to oo . Thus the relations, between the
coordinates X, F, Z of the surface and the coordinates a;, y of the part of the plane upon
which the surface of revolution is conformally represented, are
„ cos X -r^ sin A'
X = a , 1 =a ,
V y
Z=a (cos ?;+log tan \t),
where y sin t=\.
For the lower half of the surface, corresponding to the negative part of the x' , y' plane
of the original curve, the range of t is from ^tt to ; and therefore the range of y is from
1 to -f CO . The range of x, as before is from to 27r.
When in the original curve, t ranges from ir to Stt, the value of y' is complex ; the
corresponding sheet of the surface is imaginary.
When t ranges from 27r to Stt, there is a real sheet of the surface coincident with the
former real sheet. And so on, for the successive 7r-intervals of the quantity t\.
Ex. 5. In the representation of the surface of constant curvature given in the
preceding example, prove that any geodesic upon the surface becomes a circle in the x, y
plane having its centre on the axis of x.
* See my Lectures on Differential Geometry, §§ 211 — 213.
t For a further discussion of the surface and its representation upon the 2-plane, see Darboux's
TMorie generale des surfaces, t. iii, pp. 394 et seq.
From later investigations it will appear that, by other transformations, the infinite strip in
the s-plane can be represented upon different forms of areas in different planes, so that any
number of representations of the surface of revolution upon a plane can be obtained.
614 CONFORMAL REPRESENTATION [257.
257. It was pointed out (§ 254) that the conformation of surfaces is
obtained by a relation
and therefore that the conformation of planes is obtained by a relation
w=f{z\
whatever be the form of the function /, or by a relation
(/) {w, z) = 0,
whatever be the form of the function . Some examples of this conformal
representation of planes will now be considered ; in each of them the
representation is such that one point of one area corresponds to one (and
only one) point of the other.
Ex. 1. Consider the correspondence of the two planes represented by
{a-h)vfi- 2ztv + {a+b) = 0,
that is, .
7^ a-\-b
zz=(a — o)w-\ .
Let r, 6 be the coordinates of any point in the io-plane : and x, y the coordinates of any
point in the 2-plane : then
2^= (a -6)
r
2y-.
' a + b'
{a — b)r
Hence the s-curves, corresponding to circles in the H'-plane having the origin for their
common centre, are confocal ellipses, 2c being the distance between the foci, where
c^=a^ — b^: and the ^-curves, corresponding to straight lines in the w-plane passing
through the origin, are the confocal hyperbolas, a result to be expected, because the
orthogonal intersections must be maintained.
Evidently the interior of a t«-circle, of radius unity and centre the origin, is, by the
above relation, transformed into the part of the s-plane which lies outside the ellipse
x'^ja^+y^lb'^=l, the i<;-circumference being transformed into the s-ellipse.
Ux. 2. Discuss the correspondences tvz^=l, w + z^ = l.
Ex. 3. Consider the correspondence implied by the relation
2K
,_i f2K \ , , , . , ,
k ^ 11) = sn ( — z\—s,nz, where x +ty = 2 =
with the usual notation of elliptic functions. Taking vj=X+iY., we have
^" 2 (X+ {F) = sn {x' + iy')
sn x' en iy' dn iy' + sn iy' en x' dn x'
" 1 — X-2 sn^ x' sn^ iy'
Let y'= ±hK' : then sn iy' =^ ± —r=. , en iy' ■= a/ , , dn iy'=Jl-{-k^ so that
, _ i • T^N _ 1 + ^' sn x' i_ en x ' dn x'
^ ^ +'^ ^~ "^ 1 +/[• sn2.r' - ik 1 +/f^^2^' '
whence (l+^)sn^' cn^Vdn^
l+/?rsn2^'' -\^l%v?x"
and therefore Jr2+F2 = l,
257.] OF PLANES • 615
which is the curve in the i/;-plane corresponding to the hnes y' = ±^K' in the /-plane,
that is, to the lines y= ± -r-^ in the z-plane.
When v=-\ 77- and x' lies between K and — K. that is, x lies between \ir and — in-, then
T is positive and X varies from 1 to — 1 ; so that the actual curve corresponding to the
line y= — ^ is the half of the circumference on the positive side of the axis of X.
ttK'
Similarly, the actual curve corresponding to the line y= — -jy^ is the half of the circum-
ference on the negative side of that axis.
The curve hereby suggested for the ^-plane is a rectangle, with sides ^ = ± -gTr,
?/= +— -=. To obtain the zy-curve corresponding to x=\iv^ that is, to x'=K, we have
4 K
7 1 / -ir • T7A cn iw
1 cn tv^
so that F=0 and X=Jc-i -. — ^, .
an ly
Now y' varies from ^K' through to —hK': hence X varies from 1 to k^ and back
from ^* to 1. Similarly, the curve corresponding to x= —\n,
that is, to x' = — K, is part of the axis of X repeated from
— 1 to —k^ and back from -k^ to —1.
Hence the area in the w-plane, corresponding to the rect-
angle in the 3-plane, is a circle of radius unity with two diametral
slits from the circumference cut inwards, each to a distance k'^
from the centre.
The boundary of this simply connected area is the homo-
logue of the boundary of the ^-rectangle given hj x= ±W,
y= ±^^^-— : the analysis shews that the two interiors corre- ^^S- ^"•
spond*. And the sudden change in the direction of motion of the w-point at the inner
extremity of each slit, while z moves continuously along a side of the rectangle, is due to
the fact that dwjdz vanishes there, so that the inference of § 9 cannot be made at this
point. (See also Ex. 15.)
Corollary. We pass at once from the rectangle to a square, by assuming K' = 2^ ; then
k = {j2— 1)2, and the corresponding modifications are easily made.
JEx. 4. Shew that, if z=sn^(^w,k) where iv = u + iv, then the curves « = constant,
w = constant, are confocal Cartesian ovals whose equations may be written in the form
r^ - r dn (^«, k) = cn («, k), r^ -f r dn {vi, k') = cn {vi, k'),
where r and j'l denote the distances from the foci z — and z = l.
If J2 denote the distance of a point from the third focus 0=^, find the corresponding
equations connecting r, r2 ; and r-^ , ^2 .
Shew that the curves to = K,v = K' are circles, and that the outer and the inner branches
of an oval are given by lo and 2K-u, or by v and 2K'-v. (Math. Trip., Part II., 1891.)
* For details of corresponding curves in the interiors of the two areas, see Siebeck, Crelle,
t. Ivii, (1860), pp. 359—370; ib., t. lix, (1861), pp. 173—184 : Holzmiiller, treatise cited (p. 2, note),
pp. 256—263 : Cayley, Gamb. Phil. Tram., vol. xiv, (1889), pp. 484—494, Collected Mathematical
Papers, vol. xiii, pp. 9 — 19.
616
EXAMPLES OF
[257.
Ex. 5. The zf-plane is conformally represented on the s-plane by the equation
C \\—Wi
where h and c are real positive constants.
Shew that, if an area be chosen in the w-plane included within a circle, centre the
origin and radius unity, and otherwise hounded by two circles centres 1 and — 1 (so that
its whole boundary consists of four circular arcs), then the corresponding area in the
2-plane is a portion of a ring, bounded by two circles, of radii c^ and ce~'' and centre the
origin, and by two lines each passing from one circle to the other.
Prove that, when the semi-circles in the w-plane are very small, so as merely to
exclude the points 1 and —1 from the circular area and boundary, the corresponding
2-figure is the ring with a single slit along the axis of real quantities *.
Ex. 6. Consider the correspondence implied by the relation
z = c sin iv.
Taking w=X-{-iY., we have
X + iy = c sin {X + ^ T)
= c sin X cosh Y+ ic cos X sinh F,
so that X = c sin X cosh Y, y = c cos X sinh Y.
When Y is constant, then z describes the curves
- +
x^
r
=1,
c^ cosh^ Y ' c'^ sinh^ Y
which, for different values of Y, are confocal ellipses.
Now take a rectangle lying between ^^= + 577, Y=±'k. For all values of X,
cos X is positive : hence when Y= + X, y
is positive and x varies from c cosh A to
— c cosh X, that is, the half of the ellipse on
the positive side of the axis of 3/ is covered.
Let X= — ^TT : then
3/ = and X— —c cosh Y.
As Y varies from +X through to —X
along the side of the rectangle, x passes
from B to H (the focus) and back from H
to B.
When F= —X, then z describes the half of the ellipse on the negative side of the axis
of y : when X= +^7r, then ,?/ = 0, x = c cosh Y, so that z passes from A to S (a focus) and
back from 5 to J.
Hence the s-curve corresponding to the contour of the ^(J-rectangle is the ellipse
with two slits from the extremities of the major axis each to the nearer focus : the
analytical relations shew that the two interiors correspond.
Ex. 7. Consider the correspondence implied by the relation
/2K .
■ sn — sni "
V T
-(v^
From Ex. 3, it follows that the interior of a w-circle, centre the origin and radius
unity, corresponds to the interior of the ^-rectangle bounded by ^= + |^7r, 3/= ± ^-^ ,
See reference, p. 488, note.
257.] PLANE CONFOEMAL REPRESENTATION 617
provided two diametral slits be made in the w-circle along tlie axis of a; to distances
1—k^ from the circumference ; and, from Ex. 6. it follows that the same ^-rectangle is
transformed into the interior of the z-ellipse
^ f_
where a=c cosh ^-=- and 6 = c sinh — ^^ , provided two slits be made in the elliptical area
along the major axis from the curve each to the nearer focus.
Thus, by means of the rectangle, the interiors of the slit w-circle and the slit j-ellipse
are shewn to be conformal areas.
But the lines of the two slits are conformally equivalent by the above equation. For
the slit on the positive side of the axis of x extends from x=c to ^ = ccosh\, where
X=--^:^, and it has been described in both directions: we thus have
4A
z = c cosh /3,
where /3 passes from to X and back from X to 0. Hence
sin~ 1 - = sin~'i (cosh /3) = \tt + i0,
so that the corresponding iv-cnrve is given by
k -w=sn A-1
f2K0t
en —
■)
W"^'
-^
Then, when /3 assumes its values, w passes from 1 to t^ and back from X-^ to 1, that is,
w describes the circular slit on the positive side of the axis of X.
Similarly for the two slits on the negative side of the axis of real quantities. Thus
the two slits may be obliterated : and the whole interior of the ^(;-circle can be represented
on the interior of the ^-ellipse.
From the equations defining a and b, it follows that
-by -^'
in the Jacobian notation ; and c^=a^— b^.
Combining the results of Ex. 1 and Ex. 7, we have the theorem* : —
The part of the z-plane, which lies outside the ellipse x'^/a^-\-9/^lb^ = l, is transformed
into the interior of a iv-circle^ ■ of radius \inity and centre the origin^ by the relation
(a-6)w2_22w+(a + 6) = ;
and the part of the z-plane, which lies inside the same ellipse, is transformed into the interior
of the same w-circle by the relation
Jc~'^tv = sn\ '^— sin~i{3(a2_62)~2|
tvhere the Jacobian constant q luhich determines the constants of the elliptic functions, is
given by ■
fa-b\'^
^ = 1^6,) •
* Schwarz, Ges. Werke, t. ii, pp. 77, 78, 102—107, 141.
618 EXAMPLES OF [257.
Ex. 8. Investigate the equations that effect the conformal representation of the
annular region between two confocal eUipses, whose semi-axes are ao, b^ and %, 6i, upon
the annular region between two coaxal circles whose limiting points are A and B. Prove
that, if the circles cut BA produced in Pq ^-nd P\ , then the ratio (ai + bi) : (% + b^) is one
of the anharmonic ratios of the raoge {BA, PqPi). (Math. Trip., Part II., 1896.)
Ex. 9. Consider the correspondence implied by the relation
{'W+lfz=A.
When w describes a circle, of radius unity and centre the origin, then w = e** : so that,
if r and 6 be the coordinates of z, we have
- (cos ^ -z sin ^) = (1 + 6*^)2,
or -p ( cos --i sin - j = 1 + e**** = 1 + cos ^ + 1 sin 0.
s'r \ ^ ^J
Hence -— cos ^ - 1 + - sm^ - = 1,
a
that is, ?'Cos2-=l, ^
shewing that z then describes a parabola, having its focus at the origin and its latus
rectum equal to 4.
Take curves outside the parabola given by
where yx is a constant ^ 1.
so that
therefore
so that
a series of circles touching at the point X= — 1, ^^"=0, and (for ^ varying from 1 to oo )
covering the whole of the interior of the -w-circle, centre the origin and radius unity.
Hence, by means of the relation (w + 1)20=4, the exterior of the £-space bounded by
the parabola is transformed into the interior of the w-space bounded by the circle.
Ex. 10. Shew that, if z (w** — 1)2 + 4 w" = 0, the curves in the w-plane corresponding to
the real axis in the 2-plane are three arcs of circles ; and find the angles at which they cut.
Ex. 11. Prove that the infinite half-strip in the 2-plane, bomided by
0^^^27r, 0 1, that is, R<2 cos 9 ; while, if u > ^tt, we have i2> 2 cos 9.
It thus appears that the s-space lying within the parabola ?<■= Jtt, that is, r cos2i^ = l,
is transformed into the interior of a -^y-circle, centre the origin and radius unity, by means
of the relation
By the two relations* in Ex. 9 and Ex. 12, the spaces within and without the parabola
are conformally represented on the interior of a circle.
Ex. 13. Consider the relation
I — to
then, if s=.^■ + ^^/ and w = X+iY, we have
x^ty- j:2 + (1+7)2 •
When 10 describes the whole of the axis of X from - oo to + oo , so that we can take
A''=tan^, F=0, where varies from - ^ to + ^, we have .r = cos2(^, y = sin20; and
z describes the whole circumference of a circle, centre the origin and radius 1. For
internal points of this circle \—x^-y'^ is positive: it is equal to 4F-4-{2'2 + (l + 3^)^}, and
therefore the positive half of the '2<;-plane is the area conformal with the interior of the
circle, of radius unity and centre the origin, in the 2-plane.
Ex. 14. On the circle in the s-plane in the preceding example, three points ABC are
taken as the angular points of an equilateral triangle. Circles BA., AC, CB are drawn
touching OB, OA; OA, OG; OC, OB; respectively, where is the centre. Draw the
figure in the w-plane corresponding to the curvilinear triangle ABC.
Ex. 15. Again, consider a relation
^z - {c\^
\z + ic)
„, , ^ .^^ (^■2 + y2-c2)2-4c2.z;2 + 4^•c^(c2-^2_^2)
We have X-\-iY= 1 o , , — , xo^o '
{x^ + {y + cYY
so that X= ^, , , , ,212
r=
(a-2 + / - 2CX - C2) (.372 + 3/2 + 2CX - C^)
{^'2 + (y + c)2}2
4cx{G^ — x'^ — 'lf)
{x^ + {y+c)Y '
Let x = 0, so that F=0 ; then
* Schwarz, Ges. Werke, t. ii, p. 146.
620
EXAMPLES OF
[257.
As z passes from A io B (where OA = OB = c), then y changes from -c to +c, and X
changes continuously from + qo to 0.
Let x^+y^-c^ = 0, so that r=0; then
X= ^ ~^f\ .^ = _ ^Z^ = _ tan2 U,
(2c + 2j/)^ c+y 2 ;
where 3/ = c cos 6. Hence, as z describes the semi-circular
arc BCA, the angle 6 varies from to tt and X changes
from to — 00 .
(The whole axis of X is the equivalent of AOBCA ; and
at the w-origin, corresponding to B, there is no sudden
change of direction through ^tt. The result is apparently
in contradiction to § 9 : the explanation is due to the
div
Fig. 88.
fact that -^- = at B. and the inference of ^ 9 cannot be made. Similarly for A, where
dz
is infinite. See also Ex. 3.
For any point lying within the ^-semi-circle, both x and c^ — x^—y'^ are positive, so
that Y is positive. Hence by the relation
the interior of the s-semi-circle is conformally represented on the positive half of the
ic-plane.
It is easy to infer that the positive half of the ?(;-plane is the conformal equivalent of
(i) the interior of the semi-circle ACBA by the relation w--
(iii)
CBDC
BDAB
(iv) DAGD
■+ic
z + c
Z — Gj
— ic)
ST-
And, by combination with the result of Ex. 13, it follows that the relation
■-^ic.
i^
■ic
z + ic
.z^-c^ + 2cz
conformally represents the interior of the s-semi-circle ACBA on the interior of the
««;-circle, radius unity and centre the origin.
Similarly for the other cases.
Ux. 16. Shew that, by the relation
z^ + 2z-l
''z^-2z-l
the interior of the circle | ??; | = 1 is conformally represented on the interior of a semi-circle
in the 2-plane.
257.] CONFORMAL REPRESENTATION 621
Ex. 17. Find a figure in the 2:-plane, the area of which is conformally represented on
the positive half of the 'i(;-plane by
(i) ir=z-, (ii) ^'^=(^^^\ (iii) tv=z-^{l-zY.
Ex. 18. Consider the relation
tv = ae^^ :
then jr= ae~^ cos x, T= ae~y sin x.
The curves corresponding to y = constant are concentric circumferences; those corre-
sponding to .r= constant are concurrent straight lines.
As X ranges from to ^tt, both X and Y are positive ; for a given value of x between
these limits, each of them ranges from to oc , as ?/ ranges from co to — cjo . As ^ ranges
from \tt to TT, X is negative and Y is positive ; for a given value of x between these
limits, - A' and Y range from to oo , as j/ ranges from oo to — qc .
Hence the portion of the 2-plane lying between ?/= — oo, y = oo, .r = 0, x = '77, that is,
a rectangular strip of finite breadth and infinite length, is conformally represented by the
relation
on the positive half of the t<;-plane. Combining this result with that in Ex. 13, we see
that the same strip is conformally represented on the area of a ^«;-circle, centre the origin
and radius a, by means of the relation
= aie^.
w+1
Ex. 19. Find a portion of the s-plane that corresponds, luider the relation w = e*^, to
the interior of the circle
and the portion of the ?-circumference, centre the origin
and radius c.
Taking Wq and Zq as the conjugate variables, we have
c2™ + 2 0(,) ^^(,;^
L'o-
^«(?)'
so that tviV(j=-
Now if z describe the circumference of a circle, centre the origin and radius c, we have
1 •'O"
so that ?<;?fQ = c2,
shewing that lu describes the circumference of a circle, centre the origin and radius c.
To determine whether the internal area of the ^-circumference corresponds to the
internal area of the ?(;-circumference, we take zZ(f = c^ — e, where e is small. Then
^0 ( -j = <^0 ( 2o + ^ j = 00 (^O) + \ 4>o' (2o) ;
therefore ^^,^=e^ (i + ^J^) jl-^^H il-'-^^A
i 0(2) 0o(2o)J
so that the interior of the ^-circumference finds its conformal correspondent in the interior
or in the exterior of the ^^-circumference according as
«..fi^ + .,^,
4> (2) 00 (2o)
taken along the circumference.
The simplest case is that in which (^ (2) is of degree m, so that it can be resolved into
m factors, say (li{z) = A{z — a){z-^)...{z — 6): then
c^\ . fC'' \ fc
and
A (z-a)iz-l3)...{z-6)
iu =
^«^""Vi-5.)(i-|A..(i-'j.
* Cayley, Crelle, t. evil, (1891), pp. 262—277; Coll. Math. Papers, vol. xiii, pp. 191—205.
624
EXAMPLES OF
[257.
But the converse of the result obtained — that to the ?i'-circumference there corresponds
the s-circumference — is not complete unless the correspondence is (1, 1). Other ciirves
which are real — they may be, but are not necessarily, circles — and imaginary curves enter
into the complete analytical representation on the 2-plane corresponding to the ^c-circum-
ference, of centre the origin and radius c on the -w-plane.
Ex. 22. Discuss the s-curves corresponding to | ^(^ | = 1, determined by
Ex. 23. Consider the relation
We have
W — Wq =
1-n/22 '
4 {z^-z + Vf
^"~Ti {f-zf ■
(Cayley.)
27
(22-2)2 {Zi-Z^f J
The function on the right-hand side, being connected with the expressions for the six
anharmonic ratios of four points in terms of any one ratio, vanishes for
^0 ~ -I ^0
so that
tV—tVo
Hence, taking
we have
2^■^^
2o
4 (z - So) (2^0 - 1) (s + ^0 - 1) {2 (^0 - 1) -
0} (2Zo - ^0 + 1) {2 (^0 - 1) + 1}
27
2_« -12
(s^-0)2(2o2-0o;
^t; = J^+^i'', z=x-\-iy,
4 2% (^2^.^2 _ 1) (2a^ - 1) (^'2+/-2A-){(x2+,y2-.r + l)^+/
27 (x'2+y2)2(^2+^2_2^-+l)2
Hence it appears that, when 1^=0, so that iv traces the axis of real quantities in its own
plane, the s- variable traces the curves
y = 0, ^2+2/2-1=0,
2.r-l = 0, ^2^2^2_2^=:0,
that is, two straight lines and two circles in its
own plane.
In order to determine the parts of the a-plane
that correspond to the positive part of the w-plane,
it is sufficient to take Y equal to a small positive
quantity and determine the corresponding sign of
y. Let
where Y (and therefore y) is small : then, to a first
approximation,
_27 x^{x-\f
Fig. 90.
'^ 4:{2x-l){x + l)(x-2){x''--x + lf'
and the sign of y. determines whether the part on the positive or negative side of the axis
of X is to be taken.
When X <-\, ^ is negative; z lies below the axis of x. When x is in AO, so that
x>—l<0,ix\s. positive ; z lies above. When x is in OB, so that .i- > < ^, /x is negative ;
z lies below. When x is in BC, so that .r > -I < 1, /x is positive; z lies above. When x is
in CD, so that a' > 1 < 2, jn is negative; z lies below. And, lastly, when x is beyond D, so
257.] CONFORMAL REPRESENTATION 625
that ^ >2, |i is positive and z lies above the axis of real quantities. The parts are indicated
by the shading in fig. 90.
It is easy to see that w = 0, for z = P, Q; that w = l, for z = A, B, D; and that w = cc,
for z=0, C. The zero vahie of w is of triple occurrence for each of the points P and Q;
the unit value and the infinite value are of double occurrence for their respective points*.
Note. It is easy to see that figures 89 and 90 are two diflfereut stereographic projections
of the same configuration of lines on a sphere (§ 277, I., w = 3), so that the relations in
Ex. 20 and Ex. 23 may be regarded as equivalent.
Eo% 24. Find, in the same way, the curves in the 0-plane, which are the conformal
representation of the axis of X in the w-plane by the relation t
^~ 108 (2* + 2-*- 2)2 •
Ex. 25V Shew that, by the relation
the lines, .:«; = constant in the 2- plane, are transformed into a series of confocal lemniscates
in the ?{>-plane ; and that, by the relation
where c is a real positive constant greatel' than unity, the interior of a s-circle, centre
the origin and radius unity, is transformed into the interior of the lemniscate RR' = c
in the it^-plane, where R and R' are the distances of a point from the foci (1, 0) and
(-1,0). (Weber.)
258. The preceding examples | may be sufficient to indicate the kind
of coriielation between two planes or assigned portions of two planes, that is
provided in the conformal representation determined by a relation ^ {w, z) =
connecting the complex variables of the planes. We shall consider only one
more instance ; it is at once the simplest and functionally the most important
of all§. The equation, which characterises it, is linear in both variables ; and
so it can be brought into the form
az + h
w = J ,
cz + a
where a, h, c, d are constants : it is called a homographic transformation,
sometimes a homographic or a linear substitution.
Taking first the more limited form
/^
tu = - ,
z
and writing lu = i^e'®, z = re^^, fx = Pe^^*, we have
Rr^k\ % + d = 27, that is, - 7 = 7 - d,
* See Klein-Fricke, vol. i, p. 70.
t See Klein-Fricke, vol. 1, p. 75.
X Many others will be found in Holzmuller's treatise, already cited, which contains ample
references to the literature of the subject.
§ For the succeeding properties, see Klein, Math. Ann., t. xiv, pp. 120 — 124, ib., t. xxi,
pp. 170 — 173 ; Poincare, Acta Math., t. i, pp. 1 — 6 ; Klein-Fricke, Elliptische Modulfunctionen,
vol. i, pp 163 et seq. They are developed geometrically by Mobius, Ges. Werke, t. ii, pp. 189—204,
205—217, 243—314.
F. P. 40
626 HOMOGRAPHIC [258.
and therefore the new w-locus will be obtained from the old ^^-locus by
turning the plane through two right angles round the line 7 through the
origin, and inverting the displaced locus relative to the origin. The first
of these processes is a reflexion in the line o — ycao,
* do = — bdo + Ocoh + 6oada — jCoa,
7' = bbo — Ottob — dottbo + yaao.
Here 8' and 7' are real, and 6' and Oq are conjugate imaginaries ; therefore
the equation between lu and Wn represent? a circle.
Ex. 1. A circle, of radius ?• and centre at the point (e, /), in the s-plane is transformed
into a circle in the w-^\a.ne, by the homographic substitution
az + b
%o= •
cz + d'
shew that the radius of the new circle is
ad— he
where A = (cr cos /3 + e)^ + (o- sin /3 +ff - r%
d
and a, (3 are the modulus and the argument respectively o'f - . Find the coordinates of
the centre of the i(7-circle.
Ux. 2. The inverse of a point P, with regard to a circle, is Q ; and the inverse of Q,
with regard to any other circle is R. Prove that the complex variables of P and R are
connected by a homographic relation
yz + 8
Moreover, since there are three independent constants in the general
homographic transformation, they may be chosen so as to transform any three
assigned ^^-points into any three assigned w-points. And three points on a
circle uniquely determine a circle : hence any circle can be transformed into
any other circle {or into itself) by a properly chosen homographic transforma-
tion. The choice of transformation can be made in an infinite number of ways :
for three points on the circle can be chosen in an infinite number of ways.
A relation which changes the three points z^, z^, z^ into the three points
Wi, w^, Ws is evidently
{W — Wj) (Wa - W3) _ (Z — Zi) {z.2 — Z3)
(w - W2) (wi - W3) ~ {z- Z2) {Zi -Z3)'
Hence this equation, or any one of the other five forms of changing the three
points z-^, Z2, Zo, into the three points Wj, Wg, w^ in any order of correspondence,
is a homographic transformation changing the circle through ^^j, ^2, ^3 into the
circle through w-^, w^. w^.
. 40—2
628 ■ " HOMOGEAPHIC [258.
It has been seen that a transformation of the form w =f{z) does not
change angles : so that two circles cutting at any angle are transformed by
w = , into two others cutting at the same angle. Hence* a plane crescent,
of any angle, can be transformed into any other crescent, of the so.me angle.
The expression of homographic transformations can be modified, so as to
exhibit a form which is important for such transformations as are periodic.
If we assume that w and z are two points in the same plane, then there
will in general be two different points which are unaltered by the transforma-
tion ; they are called the' fixed (or double) points of the transformation.
These fixed points are evidently given by the quadratic equation
au + b
u — -J,
cu -h a
that is, cu^ — (a — d) u — b = 0.
Let the points be a and ^, and let M denote (d — ay + 46c ; then
2ca = a-d + M^, 2c/3 = a-d- M^.
If, then, the points be distinct, we have
w.— OL _ {z — a){a — ca) z — a
w-^~ {z - /3) (a - c/3) " 7^ '
a — ca a + d — M^
where K =
a-cl3 a + c? + ilf 2 '
/ 1 Y (a + df
and therefore ( JR + -?= I =
ad — bo'
The quantity K is called the midtiplier of the substitution.
If there be a 2^-curve in the plane passing through a, the w-curve which
arises from it through the linear substitution also passes through a. To find
the angle at which the ^-curve and bhe w-curve intersect, we have w = a+ Bw,
z=a+ Sz: and then
Sw = K8z,
so that the angle between the tangents to the w-cnrve and the ^^-curve is the
argument of K. Similarly, if a ^-curve pass through ^, the angle between
the tangents to the ^^-curve and the w-curve is the argument of ^.
The form of the substitution now obtained evidently admits of reapplica-
tion ; if z^ be the variable after the substitution has been applied n times, (so
that Zq = z, z^ = lu), we have
Zn-J3 z-^'
Kirchhoff, Vorlesungen ilber matheviatische Phifsik, i, p. 286.
258.] SUBSTITUTIONS 629
The condition ttat the transformation should be periodic of the 7zth order
is that Zn = z and therefore that K'^ = 1 ; hence
(a -\- dy = ^ {ad — he) cos- — ,
where s is any integer different from zero and prime to ?i; K cannot be
purely real, and, in general, M is not a real positive quantity. The
various substitutions that arise through different values of s are so related
that, if points z-^^, z^, ..., Zn be given by the continued application of one
substitution through its period, the same points are given in a different
cyclical order by the continued application of the other substitution, through
its period.
JVote The formula in the text may be regarded as giving the ?ith power of a substi-
tution. The form of the substitution obtained is equally effective for giving the ?ith root
of a substitution : all that is necessary is to express K in the form pe ^, and the nth.
root is then
^1-° 1 1
n
Ex. 1. The value of Zn has been given by Cayley in the form
{K» + i -l){az + b) + {E'' - K) {- dz + b) _
{K''+'^-l){cz + d) + {K''-K){cz-a) "
obtain this expression. ,
Ex. 2. Periodic substitutions can be applied, in connection with Kirchhoff's result
that a plane crescent can be transformed into another plane crescent of the same angle ;
the plane can be divided into a limited number of regions when the angle of the crescent
is commensurable with tt.
Let ACBDA be a circle of radius unity, having its centre at the origin : draw the
diameter AB along the axis of y. Then the semi-circle ACE can be regarded as a plane
crescent, of angle \it ; and the semi-circle ABD as another, of the same angle. Hence
they can be transformed into one another.
We can effect the transformation most simply by taking A ( = i) and 5(= — ^) as the
fixed points of the substitution, which then has the form
w — i_ jT-z — i
w + i z+i'
The line AB for the w-curve is transformed from the z-circular arc AGB: these curves
cut at an angle ^tt, which is therefore the argument of K. Considerations of symmetry
shew that the 2- point C on the axis of x can be transformed into the w-origin, so that
-1+?.'
whence K=i, so that the substitution is
iu — i_.z—i
iv + i z+i'
It is periodic of order 4, as might be expected : when simplified it takes the form
1-1-2
^ = 1—.-
630
EXAMPLES
[258.
The figure (fig. 91) shews the effect of repeated application of the substitution through
a period. The first application changes the interior of ACBA into the interior of ABDA :
by a second application, the latter area is transformed into the area on the positive side of
the axis of 3/ lying without the semi-circle ADB ; by a third application, the latter area is
e,<-
C^3\
transformed into the area on the negative side of the axis of 3/ lying without the semi-
circle ACB ; and by a fourth application, completing the period, the latter area is
transformed into the interior of ACBA, the initial area.
The other lines in the figure correspond in the respective areas.
Ex. 3. Prove that the substitution
is periodic of order six ; express it in canonical form ; and trace the geometrical eflfect of
the application of the successive powers of the substitution upon
(i) the straight line joining ?■ and -i;
(ii) the semi-circle on the last line lying to the right of the axis of y.
Ux. 4. Shew that, if the plane crescent of Ex. 2 have an angle of -tt instead of
^TT but still have +i and - i for its angular points, then the substitution
z+t
where t denotes tan ;^ , is a periodic substitution of order 2n which, by repeated appli-
2n
cation through a period to. the area of the crescent, divides the plane into 2n regions, all
but two of which must be crescent in form. Under what circumstances will all the 2ii
regions be crescent in form ?
258.] HOMOGRAPHIC SUBSTITUTIONS 631
Ex. 5. If in the plane of the complex variable z,. two circles be drawn entirely exterior
to one another, sketch the proof of the theorem, that a function of z exists which is real
at the circumferences of these circles and, exterior to the circles, is everywhere finite and
continuous save at 5=oo, where its infinite part is ^s™, A being a real assigned constant
and n an assigned positive integer.
If o, /3 be the complex arguments of the limiting points of these circles, and a, h the
modulus of - — - upon the circles surrounding a, /3 respectively, express this function
Z — a
2 h
in terms of ^u — ^c; where 10 -a z-a
634 ELLIPTIC SUBSTITUTIONS [259.
(ii) For real elliptic substitutions, a and /3 are conjugate complexes;
hence M is negative, so that
(d - of + 46c < 0,
or {d + a)' < 4 {ad - he) < 4.
The value of K, by using the relation ad — hc = \, is
K = ^ [{a + dy-2-i (a + d) [4, - {a + dffy
It is easy to see that \K\ = 1 and that its argument is cos"^ {^ (a + df -1], so
that, if this angle be denoted by cr, we have
K = e<^^
shewing that the substitution is elliptic.
It is evident that, if z describe a circle through a and /3, its centre being
therefore on the axis of x, then w also describes a circle through a and
cutting the ^■-circle at an angle cr. The two curves together make a plane
crescent of angle 4!:
we may evidently take a + d > 2. Moreover K is real and positive, shewing
that the substitution is hyperbolic.
Taking one of the fixed points for origin and denoting by / the distance
of the other, we have and / as the roots of
au + b
u = T,
cu + d
with the conditions ad — be = 1, a + d > 2. Hence h = 0, a — d = cf, ad = 1,
K = -j', then K is greater or is less than 1 according as cf is positive or is
negative. We shall take K >1 as the normal case ; and then the sub-
stitution is
a2
w = T
cz + d
with a>l> d, a + d>2, ad = l.
Ex. 1. A 2-eurve is drawn through either of the fixed points of a real hyperbolic
substitution : shew that the w-curve, into which it is changed by the substitution, touches
the z-curve. Hence shew that any 2-circle through the two fixed points of the substi-
tution is transformed into itself.
259.] ■ ELLIPTIC SUBSTITUTIONS 635
Ex. 2. Let A be a circle through the origin and the point / ; and let Cq be the other
extremity of its diameter through/. Let a real hyperbolic substitution, having the origin
and / for its fixed points, transform Cq into Ci, Cj into C2, C2 into C3, and so on : all these
points being on the circumference of A.
Shew that the radius of a circle C„, having its centre on the axis of x and passing
through c„ and the origin, is
so that C,i is the locus of all the points c„ arising through different initial circumferences
A. What is the limit towards which C„ tends as n becomes infinitely great 1
Ex. 3. Apply the inverse substitution, as in Ex. 2, to obtain the corresponding result
and the corresponding limit.
Ex. 4. Prove that a curve of finite length will meet an infinite number, or only a
finite number, of the circles (7„, according as it meets or does not meet the circle having
the line joining the common points of the substitution for diameter.
{Note. All these results are due to Poincare.)
It follows, from what precedes, that no real substitution can be loxodromic ;
for, when the multiplier of a real substitution is not real, its modulus is
unity.
It is not difficult to prove that when a substitution, with complex
coefficients a, h, c, d, is parabolic, elliptic, or hyperbolic, then a + d is
either purely real or purely imaginary. In all other cases, the substitution
is loxodromic
Any loxodromic substitution can be expressed in the form
VI — a_j^z — a
the coefficients of the quadratic determining a and ^ are generally not real,
and the multiplier K, defined by
2K = {a + dy-2- (a + d) {{a + df - 4}^,
is a complex quantity such that, if
K = pe''",
where p and w are real, then p is not equal to unity and w is not zero.
Ex. Shew that, if a, b, c, d are real or complex integers and ad- he is equal to 1 or j,
the only possible elliptic substitutions are periodic of order 2, 3, or 6 : and construct an
example of each. (Math. Trip., Part II., 1898.)
260. Further, it is important to notice one property, possessed by elliptic
substitutions and not by those of the other classes : viz. an elliptic substitution
is either periodic or infinitesimal.
Any elliptic substitution of which a and /3 are the distinct fixed points,
(they are conjugate imaginaries), can be put into the form
«; — «_„ z — a
636 INFINITESIMAL AND PERIODIC SUBSTITUTIONS [260.
where \K\ = 1: let K = e^\ Then the mth power of the substitution is
Wr». — CL z — a
Now if 6 be commensurable with 27r, so that
ej^TT = X/fM,
then, taking m = /u,, we have
z — a
that is, w^ = z,
or the substitution is jDeriodic.
But if Q be not commensurable with Itt, then, by proper choice of m, the
argument inQ can be made to differ'from an integral multiple of Itt by a very
small quantity. For we expand ^/27r as an infinite continued fraction : let
jj/g, 'p'\(i be two consecutive convergents, so that p'q — pq' = ± 1- We have
/=^ + xr<-a,whereO is constant, that is, the corresponding lines in
the ^-plane are the radii of the circle of radius unity which represents the
free surface in the w-plane. We thus have a figure in the ^-plane.
Next, introduce another complex variable ^', defined by the equation
r = iog^;
then ^' = log y + ^(/),
so that we have another figure in another plane, the ^'-plane. The circle
F = 1 in the ^-plane becomes part of the ^'-axis of imaginary quantities,
which accordingly corresponds to the free surface in the motion, given in the
w-plane by a particular line v = constant. To the radial lines = constant
in the ^-plane, correspond straight lines in the f'-plane parallel to the real
axis in that plane, that is, by general lines v = constant.
It thus is necessary to construct a relation which shall secure that a
figure- in the ■?^-plane bounded by straight lines parallel to its axis of real
quantities shall correspond to a figure in the ^'-plane bounded by straight
41—2
644 APPLICATIONS OF CONFOKMAL REPRESENTATION [261.
lines, neither necessarily nor usually parallel to the axis of real quantities in
the ^'-plane.
Finally, this result is achieved by making the figures in the w-plane and
in the ^'-plane respectively to be represented upon one and the same half of
another plane of a final complex variable t, the boundary in each representation
being transformed into the real axis in the i-plane.
We thus have a succession of planes of complex variables, conformally
represented each upon the next and therefore all upon one another. There
is the 5-plane, the original plane of motion of the fluid. There is the
w-plane, giving the velocity potential and the stream line function of the
motion. There are the ^-plane, relatively unimportant, and the ^'-plane
(where ^' = log ^), giving the magnitude and the direction of the velocity of
the fluid. And there is the i-plane, on the same positive half of which
both the variables ^' and w are represented. In these final representations
we have some relations of the types
and so, eliminating t we have a relation
that is, a relation
iogf-;7i)=/(^)'
which, on integration, gives the connection between the variables 2 and tv.
The hydrodynamical problem can thus be made to depend upon the
solution of the associated problem in conformal representation. The examples
of the latter, which have been given already, can be used to solve some of
the hydrodynamical problems which arise : one further illustration must
suffice here.
&. Imagine a fluid moving symmetrically between two parallel walls inserted into a
relatively infinite quantity of the fluid, so as to form a sort of jet coming out between
the walls. We shall assume these to have their ends in a line perpendicular to their
direction.
Then in the s-plane, we have between the walls a fluid moving along the canal sym-
metrically with respect to the middle line and outside the canal so as to supply the canal.
The boundary of the fluid is made up of the sides of the two walls in the fluid and the
double free surface within the canal, the two free surfaces being symmetrical with respect
to the middle line.
In the ^-plane, we have
so that the boundary formed by one wall up to its extremity is given by ^ = 0, that is, by
the axis of real quantities from 00 to + 1 ; then by the circumference of a circle of radius
V= 1 and centre ihe origin, representing the free surface ; and then by = 27r, that is,
by the axis of real quantities from + 1 to oo , representing the other wall.
261.] TO HYDRODYNAMICS .645
In the {"'-plane, we have
so the boundary of the configuration in the ^'-plane is
^' + irj' = ]og l+i(f)
viz. it is two lines parallel to the axis of real quantities given by (^ = and = 27r,
together with the part of the axis of imaginary quantities intercepted between these
lines.
The representation of this bounded area upon the half of the ^-plane is (§ 269, p. 673)
given by
dC'^ P .
dt (^2_i)4-'
or, adapting the scale and settling the origin, we can take
f'=2cosh-ii;.
When {■' ranges along the line (^ = from the origin to oo , < ranges from 1 to oo along its
real axis. When f ranges along the line (f) = 2ir from the axis of imaginary quantities to
infinity, t ranges from — 1 to — oo along its real axis. When ^' ranges along the axis of
imaginary quantities from 7;' = to r]' = 2Tr, t ranges from 1 to — 1 along its real axis.
Now for the ■?y-plane, let the breadth of the jet towards its issue (that is, at a great
distance from the extremities of the walls where the jet enters the canal) be 26 ; so that
one free surface is given by v = 6, and the other by v = — 6. Let the velocity potential
curve through the finite extremities of the walls be u = (). We have in general (§ 269,
p. 673)
w = M\ogt-\-N.
At the place corresponding to the extremity of the lower wall, we have
w = (d — ib ;
and at the place corresponding to the extremity of the upper wall, we have
tt; = + lb.
Hence
26,
?« = — log t — %b.
TV
Consequently, the relation between z and vj is given by the elihiination of t between the
two equations
dz
log -r^ = t' = 2 cosh" 1 1
^ dw ^
26. ^ .,
IV = — log t — ib
TT
As a matter of fact, it is more convenient to keep the two equations and not eliminate t.
Along the free stream line, we have V=\, and so
dz __ ^i
dw~ ^ '
or writing 6 = (f) + iT, we have
dz _ ei ^
dvj '
consequently along that stream line we have
ie = 2Gosh-'^t,
and therefore
t = GOSw0,
646 APPLICATIONS OF CONFORMAL REPRESENTATION [261.
where t ranges from to 1. Also
11= — log cos T! 6
■n
along the line. But u is the velocity potential ; and along the line, we have
so that we can take
llz= —s.
Thus the equation of the free stream line is
Further, we have
s = 2 — log sec \ 6.
1 _ dz _ ie
p — iq div
along the stream line, because p^ + q^=\; so
p=-cos^, y— — sin^,
and therefore 6 is the inclination to the axis of x of the tangent to the stream line.
The foregoing equation is therefore the intrinsic equation of the free stream line. It
is easy to prove that the equivalent Cartesian equations of the line are
x = 2- (sin2 1(9 - log sec h6) ]
y= -(^ — sin^) j
TV '
taking the origin at the finite extremity of the wall. As 6 ranges from to tt, ^ ranges
from to - 00 , and y ranges from to 6, that is, ?/ = & gives the asymptotic distance
between the free stream line and the nearer wall.
Similarly for the other free stream line and the other wall. Thus, as the breadth of
the issuing jet is 26, the distance between the walls is 6 + 26 + &, that is, 4&. Thus the
breadth of the issuing jet is one-half the distance between the parallel walls, a result first
stated approximately by Borda in 1766 and first proved by Helmholtz in 1868.
For a full discussion of many similar applications of the theory of conformal repre-
sentation to the irrotational motion of a liquid in two dimensions, reference may be
made to Lamb's Hydrodynamics (4th ed.), ch. iv. "Some later investigations are due
to J. G. Leathern, Phil. Trans. (1915), pp. 439 — 487, and Proc. Lond. Math. Soc, Ser. 2,
vol. xvi, (1917), pp. 140 — 149; he gives some further references.
II. Electrostatics.
C. In all the space free from electric charges in a planar electrostatic
field, the electric potential v satisfies the equation
d^v d-v _
dx^ dy-
The equipotential lines are given by
V = constant.
The component of electric force in any direction is
dv
/^
261.] TO ELECTROSTATICS 647
where dt is the increment in the positive direction ; thus the components of
electric force parallel to the positive directions of the axes are
dv dv
dx' dy'
The direction of the line of electric force at any point is given by the
equation
'^/(-s)=''2'/(-|)'
that is, by .
^r- dec — ^ dy = 0.
dy ox ^
The left-hand side is a perfect differential, because
d_ /dv\ _^(_dv\.
^y \^y) 9^ V 9^/ '
denoting it by du, we have
du _dv du dv
dx dy' dy dx'
Eliminating v, we have
9% dhi _ _
dx" dy"^ '
and the lines of electric force are given by
u = constant.
Alike from the physical properties, and from these mathematical expressions
for equipotential lines and from the lines of electric force, it follows that the
two sets of lines are orthogonal to one another at every common point.
It thus follows from former investigations (§ 8) that, if w = u + iv and
z = X + iy,
tu = some function of z =f{z).
The electric intensitv at any point is
that is, it is equal to
dw
dz
corresponding to the velocity in the hydrodynamical problem and to the
magnification in the conformation problem. Further, v = constant along
any conductor ; thus, by Coulomb's law, if a is the surface density of electrifi-
cation at any place on the plane conductor,
1 dw j
^ ~ 4!7r dzV
648 APPLICATIONS OF CONFORM AL REPRESENTATION [261.
the electric intensity being measured outwards from the surface of the
conductor.
Thus the analysis is the same throughout for the various classes of
problems. We have interpretations in the various vocabularies of the
different subjects of applied mathematics. We shall therefore deal very
briefly with a few quite simple examples.
Ex'. 1. Let
then we have the equipotential lines given by
1) = r^ sin -J 6
in "polar coordinates, with the equivalent equation
in Cartesian coordinates. They are a family of coaxial and confocal parabolas.
In particular, when v=0, we have = from x = cc to .r=0 ; and then ^ = 27r from
d; = to ^ = 00 . Thus we have an infinitely thin conductor, in the form of a straight line
stretching along the axis of real quantities from the origin to infinity. The surface
density of electrification on this conductor at a distance d from the origin is
1 7-1
~d 2.
OTT
The lines of electric force, given by
u = r2 cos 5 ^ = constant,
are represented in Cartesian coordinates by the equation
that is, they are a family of coaxial and confocal parabolas, orthogonal to the former
family.
Ex. 2. Let
w=sin~^s.
We have (Ex. 6, § 257)
A^ = sin u cosh v, y=cos u sinh i\
The equipotential lines v= constant are given by the equation
cos^ hv sin^ hv '
a family of confocal ellipses.
In particular, when v = 0, x ranges from +1 to —1 on the positive side and on the
negative side along the real axis. Thus there is an infinitely thin conductor in the fonn
of the straight line between —1 and +1 on the axis of real quantities. The surface
density of electrification on the conductor at a distance c from the middle is
The lines of electric force are given by
x"^ y^ _^
sin^ u cos^ u
that is, they are the family of confocal hyperbolas, orthogonal to the former family.
261.] TO CONDUCTION OF HEAT 649
■. 3.
Construct electrostatic interpretations of the following equations :
(i)
■iv = ism~^ z ;
(ii)
w=\ogz ;
(iii)
3 = log ?(? ;
(iv)
z =^\iw ;
(V)
w = sn ^ ;
(vi)
e^ = en w.
For a full discussion of the application of the theory of conformal representation to
general problems in electrostatics, reference should be made to Sir J. J. Thomson's Recent
Researches in Electricity and Magnetism, ch. iii ; and to the treatise by J. H. Jeans, The
Mathematical Theory of Electricity and Magnetism.
III. Conduction of Heat.
D. The same analysis can be used for the discussion of any number of
two-dimensional problems in conduction of heat when the temperature is
steady : that is, when the temperature varies from place to place in the
plane of the two dimensions, while at every place it is independent of the
time.
For purposes of illustration, we shall assume that the conductivity K is
constant. The temperature at any point will be denoted hy v; so the flux
of heat at a point perpendicular to the axis of oc is
and at a point perpendicular to the axis of y is
f'-^,y
The condition that there is neither gain nor loss of heat at any place is
that is, K being constant,
dx dy
The curves across which there is no flux of heat — they may be called lines of
flow — are given by
dx _dy
that is,
f^dx-fjy = 0.
Because
d_fdv\ _d_ fdv^
dy \dy) dx V dxj
650
APPLICATIONS OF CONFORM AL REPRESENTATION
[261.
this last expression is a perfect differential, say du ; then
du _dv du _ dv
doc dy' dy dx'
and therefore
d^u dhi _
dx^ dy-
Once more, we have the same equations as before. We know that u + iv can
be expressed as a function oi x-\- iy ; so, writing tu = u + iv and z = x -\- iy, and
taking
w=f{z\
where f is any functional form, we can interpret the analytical relation as a
result in a steady temperature problem in the conduction of heat, by taking
the lines
V = constant
as the isothermal line's, and the lines
u = constant
as the lines of flow. The total flux of heat at any point is
that is, it is
^Sf in a third plane
conformally similar to T, then S and R are also conformally similar to one
another, whatever S may be. Hence, choosing some form for 8, it will
be sufficient to investigate the question for T and that chosen form. The
simplest of closed curves is the circle, which will therefore be taken as S :
and the natural point within a circle to be taken as a point of reference is its
centre.
Two further limitations will be made. It will be assumed that the plane
surfaces are simply connected* and one-sheeted. And it will be assumed
* The conformal representation of multiply connected plane surfaces is considered by
Schottky, Crelle, t. Ixxxiii, (1877), pp. 300 — 351. Some special cases are considered by Burnside,
Lond. Math. Soc. Proc, vol. xxiv, (1893), pp. 187—206.
654 riemann's theorem [262.
that the boundary of the area T is either an analytical curve* or is made up
of portions of a finite number of analytical curves — a limitation that arises in
connection with the proof of the existence-theorem. This limitation, initially
assumed by Schwarz in his early investigations f on conformal representation
of plane surfaces, is not necessary : and Schwarz himself has shewn J that the
problem can be solved when the boundary of the area T is any closed convex
curve in one sheet. The question is, however, sufficiently general for our
purpose in the form adopted.
Then, with these limitations and assumptions, Riemann's theorem § on
the conformation of a given curve with some other curve is effectively as
follows : —
Any simply connected part of a plane hounded by a curve T can always he
conformally represented on the area of a circle, the two areas having their
elements similar to one another; the centre of the circle can be made the
homologue of any point Oo within T, and any point on the circumference of the
circle can he made the homologue of any point 0' on the boundary of T ; the
conformal representation is then uniquely and completely determinate.
263. We may evidently take the radius of the circle to be unity, for a
circle of any other radius can be obtained with similar properties merely by
constant magnification. Let w be the variable for the plane of the circle, z
the variable for the plane of the curve T ; and let
log w = t — m + ni.
Evidently n will be determined by m (save as to an additive constant), for
m + ni is a function of z : and therefore we need only to consider m.
At the centre of the circle the modulus of w is zero, that is, e'^ is zero :
hence m must he — oo for the centre of the circle, that is, for (say) z = Zo in T.
At the boundary of the circle the modulus of iv is unity, that is, e^ is
unity : hence m must be along the circumference of the circle, that is, along
the boundary of T.
Moreover, the correspondence of points is, by hypothesis, unique for th'e
areas considered : and therefore, as e'"" and n are the polar coordinates of the
point in the copy arid as m is entirely real, m is a one-valued function,
which within T is to be everywhere finite and continuous except only at
the point z^. Hence, so far as concerns m, the conditions are : —
(i) m must be the real part of some function of z :
(ii) m must be — oo at some arbitrary point z^ :
* A curve is said to be an analytical curve (§ 265) when the coordinates of any point on it
can be expressed as an analytical function (§ 34) of a real parameter,
t Crdle, t. Ixx, (1869), pp. 105—120.
t Ges. Werke, t. ii, pp. 108—132.
§ Ges. Werke, p. 40.
263.] ON CONFORMAL REPRESENTATION 655
(iii) m must be along the boundary of T :
(iv) for all points, except Zo, within T, m must be one-valued, finite and
continuous.
Now since m + ni = log w = log R-\-i%, the negatively infinite value of m
at 2^0 arises through the logarithm of a vanishing quantity ; and therefore, in
the vicinity of z^^, the condition (ii) will be satisfied by having some constant
multiple of log {z — z^) as the most important term in m + ni ; and the rest of
the expansion in the vicinity of z^ can be expressed in the form p{z — z^), an
integral series of positive powers of z — z^, because m is to be finite and
continuous. Hence, in the vicinity of ^'o, we have
log lu = m + ni = - log (z -Zf)+p{z- z^),
where \ is some constant. This includes the most general form : for the
form of any other function for m + ni is
-\og[{z-z,)g{z-z,)\+P{z-z,),
where g is any function not vanishing when z = Zq: and this form is easily
expressed in the form adopted. Hence
1
Since w is one-valued, we must have X. the reciprocal of an integer ; and
since the area bounded by T is simply connected and one-sheeted we must
have z — z^ H. one-valued function of w. Hence \ = 1 ; and therefore, in the
vicinity of ^o;
w = {z — Zq) e^^^~^'>\
a form which is not necessarily valid beyond the immediate vicinity of Zf^,
for ]} (z — Zq) might be a diverging series at the boundary. Thus, assuming
that p (z — Zo) is when z = Z(,, we have, in the immediate vicinity of Zq,
m + ni = log (z — Zq),
a form which satisfies the second of the above conditions.
It now appears that the quantity m must be determined by the con-
ditions : —
(i) it must be the real part of a function of z, that is, it must satisfy
the equation V^wi = :
(ii) along the boundary of the curve T, it must have the value zero :
(iii) at all points, except Zq, in the area bounded by T, m must be
uniform, finite and continuous : and, for points z in the
immediate vicinity of z^, it must be of the form logr, where
r is the distance from z to Zq.
656 riem-ann's theorem [263.
When m is obtained, subject to these conditions, the variable w is thence
determinate, being dependent on z in such a way as to make the area
bounded by T conformally represented on the circle in the w-plane.
264. The investigations, connected with the proof of the existence-
theorem, shewed that a function exists for any simply connected bounded
area, if it satisfy the conditions, (1) of acquiring assigned values along the
boundary, (2) of acquiring assigned infinities at specified points within the
area, (3) of being everywhere, except at these specified points, uniform, finite,
and continuous, together with its differential coefficients of the first and the
second order, (4) of satisfying V^w = everywhere in the interior, except at
the infinities. Such a function is uniquely determinate.
But the preceding conditions assigned to m are precisely the conditions
which determine uniquely the existence of the function : hence the function
m exists and is uniquely determinate. And thence the function w is
determinate.
It thus appears that any simply connected hounded area can he conformally
represented on the area of a circle, with a unique correspondence of points in
the areas, so that the centre of the circle can he made the homologue of an
internal point of the hounded area.
An assumption was made, in passing from the equation
w = {z — Zq) e'P '^"^o'
to the equation which determines the infinity of m,, viz. that, when z — Zq,
the value of ^ (^ — z^) is 0. If the value of ^ (^; — Zq) when z = Zo be some
constant other than zero, then there is no substantial change in the conditions:
instead of having the infinity of m actually equal to \og\z — Zo\, the new
condition is that m is infinite in the same way as logl^^ — ^o|, and then a
constant factor must be associated with w. A constant factor may also arise
through the circumstance that n is determined by m, save as to an additive
constant, say 7 : hence the form of w = e''^'^^'^ will be
w = A'ey^u = All.
Since displacement in the plane makes no essential change, we may take
a form tu = Au + B, where now the conformal transformation given by w is
over any circle in its plane, the one given by u being over a particular circle,
centre the origin and radius unity.
The conformation for w is derived from that for u by three operations : —
(i) displacement of the origin to the point — B/A :
(ii) magnification equal to A' :
(iii) rotation of the circle round its centre through an angle 7 :
264] DERIVATIVE FUNCTIONS REQUIRED 65T
these operations evidently make no essential change in the conformation.
If the limitation to the particular circle, centre the origin and radius 1,
be made, evidently J5 = 0, A' =1, but 7 is left arbitrary. This constant
can be determined by assigning a condition that, as the curve C has its
homologue in the circle, one particular point of C has one particular point
of the circumference for its homologue : the equation of transformation is
then completely determined.
This determination of A', B, 7 is a determination by very special con-
ditions, which are not of the essence of the conformal representation : and
therefore the apparent generality for the present case should arise in the
analysis. Now, if w = J. ?< + 5, we have
d2 \ ^ \dz)] dz \ ^ \dz)
which is the same for the two forms ; and therefore the function to he
sought is
when the area included by C is to be represented on a circle so that a given
point internal to C shall have the centime of the circle as its homologue.
The arbitrary constants, that arise when w is thence determined, are given
by special conditions as above.
Again, if the conformation be merely desired as a representation of the
2^-area bounded by the analytical curve C on the area of a circle in the
w-plane (without the specification of an internal point being the homologue
of the centre), there will be a further apparent generality in the form of the
function. From what was proved in § 258, a circle in the zt-plane is trans-
formed into a circle in the vj-plane by a substitution of the form
_ Au+ B
'^~Cu + B'
so that, if u be a special function, w will be the more general function giving
a desired conformal representation ; and, without loss of this generality, we
may assume AB — BC — 1. Using [w, z] to denote
^(\.J-^\-i
dz"
{^"^thi s('»
d /■, dw
dz
that IS, — -. 4
w ■" \w
called the Schwarzian derivative by Cayley*, we have
{w, z] = [u, z],
* Camb. Phil. Trans., vol. xiii, (1879), p. 5, Coll. Math. Papers, vol. xi, p. 148 ; for its
properties, see the memoii' just quoted, and my Treatise on Differential Equations, pp. 106 — 108.
F. F. 42
658 SOLUTION BY BELTRAMI AND CAYLEY [264.
which is the same for the two forms: and therefore the function to he
sought is
{w, z],
luhen the area included by the analytical curve C is to be conformally repre-
sented on a circle. The (three) arbitrary constants, that arise when w is
thence , determined, are obtained by special conditions.
These two remarks will be useful when the transforming equation is
being derived for particular cases, because they indicate the character of the
initial equation to be obtained : but the importance of the investigation is
the general inference that the conformal representation of an area bounded
by an analytical curve on the area of a circle is possible, though, as the proof
depends on the existence-theorem, no indication is given of the form of
the function that secures the representation.
Further, it may be remarked that it is often convenient to represent a
^-area on a w-half-plane instead of on a «^-circle as the space of reference.
This is, of course, justifiable, because there is an equation of unique trans-
formation between the circular area and the half-plane ; it has been given
(Ex. 13, § 257). Moreover, a further change, given by u' = -, , is still
possible : for, when a, b, c, d are real, this transformation changes the half-
plane into itself, and these real constants can be obtained by making points
p, q, r on the axis change into three points, say 0, 1, oo , respectively — the
transformation then being
, u—pq—r
u = — .
u — r q — p
265. Before discussing the particular forms just indicated, we shall
indicate a method for the derivation of a relation that secures conformal
representation of an area bounded by a given curve C.
Let * the curve C be an analytical curve, in the sense that the coordinates
w and y can be expressed as functions of a real parameter, say of u, so that
we have x=p (u), y = q (u) ; then
z = x + iy =p + iq = (w) ;
and the curve G is described by z, when lu moves along the axis of real
quantities in its plane.
When the equation x -\-iy = (^ {ii + iv) is resolved into two equations
involving real quantities only, of the form x — \ {u, v), y = fx {u, v), then the
eliminations of v and of u respectively lead to curves of the form
yjr (x, y, u) = 0, X (^. 2/. ^) = 0,
* Beltrami, Ann. di Mat., 2"" Ser., t. i, (1867), pp. 329—366 ; Cayley, Quart. Journ. Math.,
vol. XXV, (1891), pp. 203—226, Ooll. Math. Papers, vol. xiii, pp. 170—190; Schwarz, Ges. Werke,
t. ii, p. 150.
265.] FOR ANALYTICAL CURVES 659
which are orthogonal trajectories of one another when u and v are treated as
parameters. Evidently x {^> y> 0) = is the equation of C : also
So far as the representation of the area bounded by (7 on a half-plane is
concerned, we can replace w by an arbitrary function of Z{=X + iY) with
real coefficients : for then, when Y= 0, we have w=f(X) and
^=p{f(X)], y = q[f{X)],
which lead to the equation of G as before, for all values of /. This arbi-
trariness in character is merely a repetition of the arbitrariness left in
Gauss's solution of the original problem.
Now let the w-plane be divided into infinitesimal squares with sides
parallel and perpendicular to the axis of real quantities. Then the area
bounded by Cis similarly divided, though, as the magnification is not every-
where the same, the squares into which the area is divided are not equal to
one another. The successive lines parallel to the axis of u are homologous
with successive curves in the area, the one nearest to that axis being the
curve consecutive to G. Similarly, if the ^-plane be divided.
Conversely, if a curve consecutive to G, say G', be arbitrarily chosen, then
the space of infinitesimal breadth between G and G' can be divided up into
infinitesimal squares. Suppose the normal to C at a point L meet G' in L' :
along G take LM= LL', and let the normal to G at M meet G' in M'\ along
G take MN= MM', and let the normal to C at iV meet G' in N': and so on.
Proceeding from G' with L'M', M'N', ... as sides of infinitesimal squares, we
can obtain the next consecutive curve G", and so on; the whole area bounded
by G may then be divided up into an infinitude of squares. It thus appears
that the arbitrary choice of a curve consecutive to G completely determines
the division of the whole area into infinitesimal squares, that is, it is a
geometrical equivalent of the analytical assumption of a functional form
which, once made, determines the whole division.
Next, we shall shew how the form / of the function can be determined
so as to make the curve consecutive to C a given curve. As above, the
curve G is given by the elimination of a (real) parameter between
x = p (u), y=^q{u);
and the representation is obtained by taking
x + iy = z=p{w) + iq (w)=p [f{Z)} + iq [f{Z)].
Let the arbitrarily assumed curve G', consecutive to G, be given by the
elimination of a (real) parameter 6 between
x=p + eP, y = q + eQ,
where p, P, q, Q are functions with real coefficients, and e is an infinitesimal
constant : the form of/ has to be determined so that the curve corresponding
42—2
660 CONFORMATION OF AREA [265.
to an infinitesimal value of Y is the curve C Taking u=f{X), where u
and X are real, we have, for the infinitesimal value of T,
^ + ii/=p{fi^)]+n{fm
so that ^ = p — y-y^^, y = 9+^'yYP'
dashes denoting difi'erentiation with regard to u. This is to be the same
as the curve C, given by the equations
x=p + eP, y = q + eQ.
Hence the (real) parameter 6 in the latter differs from u only by an infini-
tesimal quantity : let it be ti — fi, so that we have
x=p — jxp' + eP, y = o — fJ^(l' + eQ,
the terms involving products of e and yu. being neglected, because they are
of at least the second order. Hence
-^p' + eP = -Y^q', -^q^+eQ=Y^p';
whence /^ (p'' + q'^) = e (Pp' + Qq'),
and e{p'Q-q'P)=Y^{f^ + (^^r.
Now e is a real infinitesimal constant, as is also F for the present purpose :
so that we may take 6 = AY, where J. is a finite real constant : and A may
have any value assigned to it, because variations in the assumed value
merely correspond to constant magnification of the ^-plane, which makes
no difference to the division of the area bounded by C Thus
A{p'Q-q'P)=^{p'-^ + q%
and therefore AX = I —rp^ 77:; du,
J pQ-qP
the inversion of which gives u=f{X) and therefore w =f{Z), the form
required.
Also we have fi = A Y?K^^^ ,
p^ + q^
shewing that, if the point oc=p + eP, y = q + eQ on C lie on the normal to C
a,t x=p, y = q, the parameters in the two pairs of equations are the same;
the more general case is, of course, that in which the typical point on C is in
* Beltrami obtains this result more directly from the geometry by assigning as a condition
that the normal distance between the curves is equal to the arc given by du : I.e., (p. 658, note),
p. 343.
265.] BOUNDED BY ANALYTICAL CURVE 661
the vicinity of C. And it is easy to prove that the normal distance between
the curves at the point in consideration is
„ ds
dX'
where ds is an arc measured along the curve C.
Ex. 1. As an illustration* let Cbe an ellipse x^ I a?' + y'^ jl)^ = \ and let C be an interior
confocal ellipse of semi-axes a — n, 6-/3, where a and /3 are infinitesimally small ; so that,
since
(a-a)2-(6-^)2 = a2-62 = c2,
we have aa=bj3—C€ say; then the semi-axes of C are a — e, b-^e. We have
a b
p = a cos u, q=bsinu,
c
b'
P= — COS.U, ^=-^sini«,
so that AX=\ — du=~u,
J c c '
or, taking A= — , we have X=u and therefore Z=w. Hence the equation of transform-
ation is
z=x + iy = a cos Z+ib sinZ;
or, if a = c cosh Yq, 6 = csinh Fq, and if Y' denote Fq- Y, the equation is
z = c cos {X+iY') = c cos Z'.
The curves, corresponding to parallels to the axes, are the double system of confocal
conies.
Ex. 2. When the curve C is a parabola, with the origin as focus and the axis of real
quantities as its axis, and C is an external confocal coaxial parabola, the relation is
z=a{Z+if;
substantially the same relation as in Ex. 9, § 257.
Ex. 3. When C is a circle with its centre on the axis of real quantities and C is
an interior circle, having its centre also on the axis but not coinciding with that of C, the
circles being such that the axis of imaginary quantities is their radical axis, the relation
can be taken in the form
z=c tan Z. (Beltrami ; Cayley.)
. Note. Although, in the examples just considered, the successive curves
G ultimately converge to a curve of zero area (either a point or a line), so
that the whole of the included area is transformed, yet this convergence
is not always a possibility, when a consecutive to G is assigned arbitrarily.
There will then be a limit to the ultimate curve of the series, so that the
representation ceases to be effective beyond that limit. The limitation
may arise, either through the occurrence of zero or of infinite values of -^^
for areas and not merely for isolated points, or through the occurrence of
branch-points for the transforming function. In either case, the uniqueness
of the representation ceases.
* Beltrami, I.e., (p. 658, note), p. 344 ; Cayley, (ib.), p. 206.
662 EXAMPLES • [265.
Ex. 4. Consider the area, bounded by the cardioid
r=2a (1+cos^);
then we can take
x=p = 'ia (l + cos«) cos u, y = q='2a {\ + cos u) sin u,
where evidently u = d along the curve. Let the consecutive curve be given by
^= -a6 + 2a(l + e) i\ + GOsu')cosu', y = 2a(l + e) (1 + cos M')sin u',
so that, to determine X, we assume P= — a+2a (1 + cosm) cos^f, ^ = 2a (1+cosm) sinw,
for m' — 2«= -/a a small quantity.
We have p"^ + q''^ = 16a^ cos^ ^i,
q'P—p'Q=12a? cos2 ■!«.,
p'P+q'Q= — 2a^ sin u ;
and then, proceeding as before and choosing A of the text as equal to — |^, (which implies
that e is negative and therefore that the interior area is taken), we find
X=u,
therefore Z= w. Thus the cardioid itself and the consecutive curves are given by
iZ
zz=p + iq = 2a{l+cos Z)e .
To trace the curves, corresponding to lines parallel to the axes of X and Y, we have
'^^'' = 2cos^Zei'^,
Hence, multiplying, we have
and, dividing, we have
(f)* = 2c„.K.«-i^
r = 4:ae {cos ^ Z cos ^ Zq)
= 2ae (cosh T+ cos X) ;
iX cos^Z
cos l^o'
,•, , . i{X-e)_cos^X cosh^T+isin ^Xsmh^Y
^ ^^' ^ ~ cos i X cosh I r- i sin ^X sinh ^Y'
and therefore
Moreover, we have
tan ^{X-0) = tan | X tan h i Y.
dz ^ . iz,, , iZ.
-^=2aie (1 + e ),
which vanishes when Z=Tr (2% + l), that is, at the point X={'2,n+\)Tt, F=0; whence the
cusp of the cardioid is a singularity in the rej)resentation.
When F=0, then X=^d and r=2a (1 + cos(9), which is the cardioid; when F is very
small and is expressed in circular measure, then
tan|(Z-^)=irtaniX,
or .r=(9+rtani^,
so that r =2a {I + cos 6) -AaY.
It is easy to verify that
6=u' + \Yisji\u',
agreeing with the former result.
The relation may be taken in the form
(«/a)* = 2 cos lZe^^^ = e'^+l,
265.]
OF ANALYTICAL CURVES
663
which shews that z = a is a branch -point for Z. Two different paths from any point to a
point P, which together enclose a, give different values of Z at P. Hence the representa-
tion ceases to be effective for any area that includes the point a.
Consider a strip of the Z-plane between the lines F=0, Y=+
668 RECTILINEAR POLYGON [267.
All kinds of points on the boundary of the ty-polygon have been con-
sidered, corresponding to points on the axis of x.
We now consider points in the interior. If lu' be such an interior point
and z be the corresponding ^--point, then
w — iv' = {z — z') S {z — z'),
where S does not vanish for z = z' , because at every point -7 - must be different
from zero : for otherwise the magnification from a part of the ^•-plane to a
part in the interior of the polygon would be zero and the representation
would be ineffective.
Now in the present case, just as in the first case suggested in § 264, it is
manifest that, if a particular function w' give a required representation, then
Aw'^-B, where |^| = 1, will give the same w-polygon displaced to a new
origin and turned through an angle a, = arg. A, that is, no change will be made
in the size or in the shape of the polygon, its position and orientation in the
w-plane not being essential. Hence the function to be obtained may be
expected to occur in the form lu = Aiu' + B ; hence
dw _ . dw'
dz dz '
and therefore log -y- = log e^" + log -j— ,
,, , d L dw\ d /, dw'\
BO that £rs&j=sr»3rj-
Thus, in representing a figure hounded hy straight lines, the function to he
ohtained is
Now in the vicinity of a boundary-point /3, not being an angular point
and corresponding to a finite value of z, we have
w-l3 = e^(-^+^\z-b)P(z-h),
and therefore Z= Pi{z — b),
having z = b for an ordinary point.
For a boundary-point /3', not being an angular point and corresponding
to an infinite value of z on the real axis, we have
w-/3' = e*'(-+^'> IqI]),
2 ] /1\
and therefore Z — h — Qi ( - 1 ,
z z'- \z )
where Qi is not zero for ^ = 00 . Thus Z vanishes for such a point.
267.] REPRESENTED ON A HALF-PLANE 669
In the vicinity of an angular point 7, corresponding to a finite point c on
the real axis, we have
and therefore Z = \- B.,(z — c),
Z — G
where i^^ has z-=g for an ordinary point.
In the vicinity of an angular point 7', corresponding to a place at
infinity on the real axis, we have
z z- \zj
where R^ is not zero when z = ao . Thus Z vanishes at such a point.
Lastly, for a point lu' in the interior of the polygon, we have
w-iu'={z-z')S{z-z'),
and therefore Z = Si(z — /),
having z = z' for an ordinary?- point.
Hence Z, considered as a function of z, has the- following properties: —
It is an analytical function of z, real for all real values of its argument,
and zero when a; is infinite :
It has a finite number of accidental singularities each of the first order
and all of them isolated points on the axis of-^: and at all other
points on one side of the plane it is uniform, finite and continuous,
having (except at the singularities) real continuous values for real
continuous values of its argument.
The function Z can therefore be continued across the axis of x, conjugate
values of the function corresponding to conjugate values of the variable: and
its properties make it, by § 48, a rational meromorphic function of z.
Let a,h,c,...,l be the points (all in the finite part of the plane) on the
axis of X corresponding to the angular points of the polygon, and let
OTT, /StT, ryir, ..., XlT
be the internal angles of the polygon at the respective points.
The residues at these points are a— 1, ;8-l,7 — 1,...,A,— 1 respectively ;
so, after § 48, we consider the function W given by
[z — a z — z - c z — L
It is finite everywhere in the finite part of the plane z.
670 RECTILINEAR POLYGON [267.
If no angular point of the polygon corresponds to ^^=00, then for large
values of z we have
where T\-\ is finite for large values of z. But
S (tt — om) = sum of the external angles of the polygon
= 27r,
so that 2 (a - 1) = - 2.
Thus W is zero of the second order when z = cc .
If an angular point of the polygon corresponds to z = 00 , then for large
values of z we have
n.=_'i±.i+\ij.(i)_l2(«-i)-lr(i),
z z^ \zj z z^ \zj
where T{-\ is finite for large values of z. But now
(tt — /i7r) + S (tt - air) = sum of the external angles of the polygon
• = - 27r,
so that 1 - /i + 2 (1 - a) = - 2,
that is, /A + l + 2(a-l) = 0.
Thus W is. zero of the second order when z — ^ .
Consequently W has no pole anywhere in the ^r-plane ; so it is a constant.
At infinity, W is zero, so that this constant is zero. Hence we have ,
d L fdw\] _ « — 1 A. — 1
dz\ ° \dz )\ z — a '" z — i
dz
and each of the quantities a, /3, ..., X- is less than 2. This equation*, when
integrated, gives
w = GJ(z - ay-' (z - by-' ...(z- ly-' dz + C,
where G and C are arbitrary constants, determinable from the position of
the polygon •(-.
• It is to be noted that, when no angular point of the w-polygon has its
homologue at infinity, the relation
2 («-!) = -2
is satisfied ; and that this relation does not hold when an angular point of
the w-polygon has its homologue at infinity.
The final expression for w in terms of z is commonly called the Schwarz-
Christoffel transformation.
* This relation is made the basis of some interesting applications in hydrodynamics, by Michell,
Phil. Trans., (1890), pp. 389 — 43 L See also the authorities quoted in the " Note on some appli-
cations of couformal representation to mathematical physics," pp. 639 — 652, ante.
t This result was obtained independently by Christoffel and by Schwarz: I.e., p. 666, note.
268.] TRIANGLE ON A HALF-PLANE 671
268. It may be remarked, first, that any three of the real quantities
a, h, c, . . . , I can be chosen arbitrarily, subject to the restrictions that the
points a, h, c, ..., I follow in the same order along the axis of w as the angular
points of the polygon and that no one of the remaining points passes to
infinity. For if three definite points, say a, b, c, have been chosen, they can,
. by a real substitution
where p, q, r, s are real quantities satisfying ps — qr = 1, be changed into
other three, say a, b', c' : and then, substituting
and using the relation S(a — 1)= — 2,
we have tu = TJ(^-ay-'(^-by-'...{^-iy-'d!;+C\
where F is a new constant. By the real substitution, the axis of real
quantities is preserved : and thus the new form equally effects the con-
formal representation of the polygon.
But, secondly, it is to be remarked that when three of the points on the
axis of a; are thus chosen, the remainder are then determinate in terms of
them and of the constants of the polygon.
269. The simplest example is that of a triangle of angles utt, /Sir, j-jr, so
that
a + /3 + y = l.
Then a particular function determining the conformal representation of this
w-triangle on the half 2^-plane is
_ r dz
^^ ~ j (^ - a)i- {z - by-^ (z - cf-y '
dz
so that T- = (^ - «y~" (^ - by~^ (^ - oy-^ >
aw
a differential equation of the first order*.
For general values of a, yS, 7, which are rational fractions, the integral-
function w is an Abelian transcendent of some class which is greater than 1 :
and then, after §§ 110, 239, z is no longer a definite function of w, and the
path of integration must be specified for complete definition of the function.
If a = 0, the only instance when the integral is a uniform function of w
is when yS = i, 7 = i : and then the function is simply-periodic. In such
a case, the w-figure is a strip of the plane of finite breadth, extending in one
* Equations of this type are discussed in my Theory of Differential Equations, Part 11, vol. ii ;
in particular, see Chapters ix, x. The cases when the integrals are uniform functions, either
algebraic, simply-periodic, or doubly-pei iodic, are discussed in § 137 of the volume quoted.
672 SQUARE ON A CIRCLE [269.
direction to infinity and terminated in the finite part of the plane by a straight
line perpendicular to the direction of infinite extension.
Ex. Discuss the case when a = 0, i3=0, 7=1, pointing out the relation between the
half of one plane and a strip in the other.
If no one of the quantities a, ^, 7 be zero, then on account of the condition
aJ^ ^ -\. ^ = 1, the only cases when the integral gives ^^ as a uniform function
of IV are as follows : —
(I) a = i, /3 = J, 7=3; an equilateral triangle :
(II) 0L = \, /3 = \, 7 = i; an isosceles right-angled triangle :
(III) a = i, ^ = i, 7 = i; ^ right-angled triangle, with one angle equal
to ^TT.
In each of these cases z is a, uniform doubly-periodic function of w ; and
after arranging the constants of integration, the respective relations are
(I) -j = ^'(^o),
where the invariants g2, gz are = 0, 4 ; .
(H) ^=r(^).
where the invariants g^, gz are = — 4, ;
(HI) 1^. = ^'W'
where the invariants g^,, gz are 0, 4.
The integral expressions for these cases were given by Love*, who also
discussed a further case, (due to Schwarz), in which z occurs as a two-valued
doubly-periodic function of w. The triangle is then isosceles with an angle
of f TT, the values of a, /3, 7 being a = f, ;8 = ^,7 = |-; the relation is
where the invariants g^ and g.^ are 0, 4.
The example next in point of simplicity is furnished by a quadrilateral,
in particular by a rectangle : then
a = ;S = 7 = 8 = i.
When no one of the homologues of the angular points is at infinity on
the real axis in the ^-plane, the general form is
'w=j[{z—a){z — h){z — c){z — d)\~^dz,
so that 2^ is a doubly-periodic function oi z.
* Amer. Journ. of Math., vol. xi, (1889), pp. 158—171.
269.] SQUARE ON A CIRCLE 673
When one (but only one) of the homologues of the angular points is at
infinity, the form is
w=J[(2-a)(z- b) {z - c)]~^dz,
so that again ^ is a doubly-periodic function of z.
When two of the homologues of the angular points are at infinity — and
there cannot be more than two — then after the explanations of § 267, the
form of relation is
dz\ dz J z — b z — c'
We shall consider the last form at once. There are two alternatives
according as b and c are distinct or as they coincide. When they are distinct,
we take 6 = + l, c = — 1; when they coincide, we take 6 = c = 0.
For the first case, we easily find
w = A \ — : + B
-' {Z' - 1)2
= A cosh~^ z + B.
Dropping the constant B (which only affects the origin in the w-plane),
we have
w
z = cosh , ,
A
and we shall take A real. Thus
u V . , u . V
^ cos -^ , 2/ = smh J sm ^ ;
the upper half of the ^-plane is conformally represented upon the interior of
a strip in the w-plane bounded by the lines v = 0, v = 7rA, and the part of
u = between these two lines.
For the second case,
we have
d A dw\
dz\ ^^ dz) ~
1
z '
and so
we easily find
w = A log z 4-
■B.
Again
dropping the constant B and taking
A real,
we
have
u
X = e^ cos -T , y =
= e^ sin
V
the upper half of the ^;-plane is conformally represented upon the interior of
the strip in the w-plane bounded by the lines v = 0, v = irA.
Passing now to the general form, we shall first suppose that the rectangle
is a square. We choose oo , 1, as points on the axis of x corresponding to
three of the angular points in order. The symmetry of the figure then
F. F. 43
674
RECTANGLE ON
[269.
enables us to choose — 1 as the point on the axis of x corresponding to the
remaining angular point. As the homologue of one angular point is at
infinity, and as — 1, 0, 1 are the homologues of the other three, we have the
relation between w and z in the form
w = CJ{z {z - 1) (^ + V)\-^dz + G'
dz
=
-T + C,
{z(z^-l)}^
C and C being dependent upon the position and the magnitude of the
w-square.
Again, the half ^-plane is transformed (§ 257, Ex. 13) into the interior of
a ^-circle, of radius 1 and centre the origin, by the relation
I + Z
Then except as to a constant factor, which can be absorbed in C, the integral
in IV changes to
dZ
so that, by the relation
W =
(1 - Z'f
dZ
(1 _ z^y
the interior of a ^-circle, centre the origin and radius 1, is the conformal
representation of the interior of some
square in the TT-plane. Denoting by
L the integral 1 , so that 2L
is the length of adiagonal, the angular
points of the square are D, A, B, G
on the axes of reference : and these
become d, a, h, c on the circumference
of the circle. They correspond to
representation on the half-plane;
Ex. Shew that the area outside a square in the w-plane can be contbrmally repre-
sented on the interior of a circle in the s-plane, centre the origin and radius unity, by the
equation
Fig. 94.
1, 0, 1, GO on the axis of x in the
J 1 2"
the 2-origin corresponding to the infinitely distant part of the ?6'-plane. (Schwarz.)
Secondly, let the rectangle have unequal sides. Then the symmetry of
the figure justifies the choice of j, 1,— 1, — ^ as four points on the axis of x
269.] A HALF-PLANE 675
corresponding to the angular points of the rectangle when it is represented
on the half-plane. We thus have
= g!" {(1 - z') (1 - A;V^)|- idz+ G\
w
J
If the rectangle be taken so that its angular points are a, a + 2bi, -a + 2bi,
- a in order, these corresponding to 1, ^ , - r , - 1 respectively, then we have
so that the relation is
and then
0:
= C',
a-
= CK,
a
+ 2bi :
= C(K + iK');
w
a
K =
'Jo
{(1-.
K'
K~
q = e
_2b
a '
27r6
a
whence
where q is the usual Jacobian constant : this equation determines the relation
between the shape of the rectangle and the magnitude of k.
In the particular case when the rectangle is a square, we have b = a and
K' - 1 -
so ^ = e-2^ or -^ = 2 : and therefore * ^^ - 3 - VS or y = 3 -f- V8. The differ-
ence from the preceding representation of the square is that, there, the point
z = i was the homologue of the centre of the square, whereas now, as may
easily be proved, the point z = i{\l2 -[- 1) is the homologue of the centre.
Ex. Discuss the transformation
/"2 1 1
tO=\ -,(1-202 COS a -I- 3*) "2(^0^
shewing that the perimeter of a rectangle in the w-plane is transformed into the circum-
ference of a circle in the z-plane. Discuss also the correspondence of the areas bounded
by these perimeters.
But in the case of a quadrilateral, in which such symmetrical forms are
obviously not possible, and, in the case of any convex polygon, only three points
can be taken arbitrarily on the axis of x : the most natural three points to take
are 0, 1, x for three successive points. The values for the remaining points
must be determined before the representation can be considered definite.
* This is derived at once by means of the quadric transformation in elliptic functions.
43—2
676 QUADRILATERAL [269.
Thus in the case of a quadrilateral, taking oo , 0, 1 as the homologues of
D, A, B respectively and - as the homologue of C, c
(where yu.00, a' + 1 -y' = 2 -y- a- /3 = S > 0,
SO that, as jLt < 1, the definite integral is finite at all the critical points.
* For the analytical relations in reference to the definite integrals, see Goursat, " Sur
r^quation differentielle lineaire qui admet pour integrale la s^rie hypergeometrique," Ann.
de VEc. Norm. Sup., 2°^= Ser., t. x, (1881), Suppl., pp. 3—142; and for the relations between the
hypergeometric series, see my Treatise on Differential Equations, 4th ed., pp. 211 — 230, 290 — 293,
the notation of which is here adopted.
269.]
We have
DETERMINATION OF CONSTANT
677
/>^=-j«ji&;=«^K ,■„.,,)
r(a)r(/3)
Ti;
,r.(ff-i) r-(g)r(y) _
r(/3+y) '^*'
xi^U'-/3', l-,3', y'-a'-/3' + l,
_ -..■(i3+v)r(y)r(S)
Xi^(^^'-y' + l, 1-a', 2-y', -^
r(y+8)
^T-'I^2•
Hence
Now, if
r(a)r(^) ^ ^.•(/3-i)r O)r(y) x. , -«(0+y)r^yOr^) ..
^^ '^ r(a+/3) -^^-' r(y+S) ^*+' TFPsy ■
J/l=-
n(y-i)n(-a')n(-/y)
r(a+^)r(y)r(i-a)
n(l-y')n(y'-a'-l)n(y'-/3'-l) T (y + S) T (1 -8) T (/3) '
^,^ ^ n(-aOn(-^') _ r(y)r(i-«)
n(y'-a'-i3')n(-y') r(/3 + y)r(y-i-S-i)' ■
then ri = Af^Yo+N^Yi.
Substituting, we have
*L^ ^ r(a+^) r(^+y) _
^L r(y + S) ^ ^ r(a + i3) i_
By using properties of the Gamma functions, the coefficient of Y^ can be proved equal to
e"'" r((3)r(y),, . , ^, - ^ -, e""siny7r r (^) r (y)
asmoTT r(/3 + y) ^ ^ ' ' asmoTr r(/3 + y) '
and the coefficient of Y.2 can be proved* equal to
e"° r(y)r(g) ,^ . . _^^^ , w«_L. ^ ) e"Siny7r,r(y)r(8)
— ^ '/ . ^N {asm (a + j3 + y) 7!--a sm (/S+y) 7r}= ^ ^— h —rj-^ — ^ .
asmaiv r(y + 8) ^ ^ 'MT-// \t-^ r/ j a sm ott r(y + S)
Moreover
l-y
SO that
72=^20=^4 = / ^(l-/^f "" ^ F{l-a', 1-0', 2-y', m},
"^>{l-a', 1-/3', y'-a'-/3' + l, 1-/.}:
Fa i^{y, 1-a, y + 8, /x} .
^4=y24=J/8 = /^^ ^ (1-M
Fi /'{y, 1-a, y+ft 1-;^}'
and therefore an equation to determine /i is
F{y, l-g, y + d, ^} ^C V{^)V{y + b)
F{y, 1-a, y + /3, l-^a} 6 T (S) T (y+/3) '
* In reducing the coeflBcients to these forms, limiting cases (such as ;8 + y=l) of the quadri-
lateral are excluded.
678 LIMITING CASE OF POLYGON [269.
Ex. 1. A convex quadrilateral in the w-plane has its sides AB, BC equal to unity, its
angles A and C each equal to air, and its angle B equal to /Stt ; prove that it can be
conformally represented upon the positive half of the s-plane by the relation
-1/1 ^.•iNa-i j„_ / -^-1/1 „2n'^~i
"l
2
Ts
_2
(1-
-^") » dx=
/ ^1-
-s") ^dz.
^
/o
x^-\l-x'f 'dx= z^ \\-z^T dz.
(Math. Trip., Part II., 1895.)
Ex. 2. A regular polygon of n sides, in the w-plane, has its centre at the origin
and one angular point on the axis of real quantities at a distance unity from the origin.
Shew that its interior is conformally represented on the interior of a circle, of radius
unity and centre the origin, in the 2-plane by means of the relation
(Schwarz.)
Ex. 3. A plane non-reentrant hexagon ABCDEF is symmetrical about its diagonal
AD: prove that the enclosed area can be conformally represented upon a half -plane,
so that the angular points correspond to points co , — 1, — /x, 0, /x, 1 on the real axis,
provided /x is determined by the equation
]_
where the variable 6 is real throughout the two ranges of integration, b and c are the
sides AB and BC respectively, and air, /Stt, yn are the internal angles A, B, G respectively,
(Trinity Fellowship, 1898.)
270. It is natural to consider the form which the relation assumes when
we pass from the convex polygon to a convex curve, by making the number
of sides of the polygon increase without limit. The external angle between
two consecutive tangents being denoted by d\lr, and the internal angle of the
polygon at the point of intersection of the tangents being ^tt, we have
TT — ^TT = dyjr,
dyl/'
so that ^—1 — .
TT
Let X be the point on the axis of real quantities, which corresponds to this
angular point of the polygon ; then the limiting form of the relation
d /, dw\ _ n;> a — 1
dz\°dzj 2 — a
d (. dw\ 1 i dylr
dz \ dz] TT J z — x^
where x is the point on the real axis in the ^-plane corresponding to the
point on the tu-cwxve at which the tangent makes an angle i/r with some
fixed line, and the integral extends round the curve, which is supposed to be
simple (that is, without singular points) and everywhere convex.
270.] AS A CONVEX CURVE 679
The disadvantage of the form is that x is not known as a function of -v/r ;
and its chief use is to construct curves such that the contour is conformally
represented, according to any assigned law, along the axis of real quantities
in the ^-plane. The utility of the form is thus limited : the relation is not
available for the construction of a function by which a given convex area in
the w-plane can be conformally represented on the half of the 2^-plane*.
Ex. Let A' = tan ■^■v//- : then taking the integral from — tt to +7r, we have
log
dz\ ^ dz J TT j _^^-tan|■v//■
7r 7 _i„3 — tan
The integral on the right-hand side is
p'^ d(i> _ fo
j 2 - tan (^ j 1
z + tan (f)
d(t>
/ z^ — tan^ (p
and therefore
d^ / ^\ _ _ _2_
dz\^dz)~ z-V
which, on further integration, leads to the ordinary expression for a circle on a half-
plane.
271. In regard to the conformal representation on the half of the
^-plane of figures in the 'Zt'-plane bounded by circular arcs, we proceed-f-
in a manner similar to that adopted for the conformal representation of
rectilinear polygons.
* See Christoffel, Gott. Naehr., (1870), pp. 283—298.
t For the succeeding investigations the following authorities may be consulted : —
Schwarz, Ges. Werke, t. ii, pp. 78—80, 221—259.
Cayley, Camb. Phil. Trans., vol. xiii, (1879), pp. 5—68; Coll. Math. Papers, vol. xi,
pp. 148—216.
Klein, Vorlesungen ilber das Ikosaeder, Section I., and particularly pp. 77, 78.
Darboux, Theorie generale des surfaces, t. i, (2nd ed.), pp. 512 — 526.
Klein-Fricke, Theorie der elliptischen Modulfunctionen, t. i, pp. 93 — 114.
Goursat, I.e., p. 676, note.
Schonflies, 3Iath. Ann., t. xlii, (1893), pp. 377—408, ib., t. xliv, (1894), pp. 105—124.
680 SCHWARZIAN DERIVATIVE [271.
It is manifest that, if u=f{z) determine a eonformal representation on
the 2^-plane of a w-polygon bounded by circular arcs and having assigned
angles, then
Au + B
Cu + D'
where A, B, C, D may be taken subject to the condition AD — BG = 1, will
represent on the half ^-plane another such polygon with the same assigned
angles : for the homographic transformation, preserving angles unchanged,
changes circles into circles or occasionally into straight lines. Hence, as
in § 264, when the transforming function is being obtained, it is to be expected
that it will be such as to admit of this apparent generality : and therefore,
since
[w, z] = [u, z],
where [w, z\ is the Schwarzian derivative, it follows that, in obtaining the
eonformal representation of a figure bounded by circidar arcs, the function to
be constructed is
We proceed, as in the case of the rectilinear polygon, to find the form of
the appropriate function in the vicinity of points of various
kinds. But one immediate simplification is possible, which
enables us to use some of the earlier results.
Let C be an angular point, CA and CB two circular
arcs, one of which may be a straight line : if both were
straight lines, the modification would be unnecessary. In-
vert the figure with regard to the other point of intersection
of CA and GB : the two circles invert into straight lines cutting at the same
angle /jltt. Take the reflexion of the inverted figure in the axis of imaginary
quantities : and make any displacement parallel to the axis of real quantities :
if W be the new variable, the relation between lo and W is of the form
aW + b
cW + d~'"'
rhere ad -
-bc = l; and therefore
{ W, z] = [w, z
Consider the function for the T^-plane. Let T be the point corresponding
to G, an angular point of the polygon, having ^r = c as its homologue on the
axis of x, account being taken of the possibility of having c = x ; let /3 be any
point on either of the straight lines corresponding to a point on the contour
of the polygon not an angular point, having z = b as its homologue on the
axis of x. If a contour point not an angular point have ^ = oo as its
homologue on the axis, denote it by /3'.
271.] FOR EEPRESENTATION ON A CIRCLE 681
Then for the vicinity of ^, we have (as in § 267) a relation of the form
F- /3 = e*>+«) (z-b)P(z-b);
then
so that
dW
log -7— = const. 4- log Pi (z — b),
{W,z]=P,(z-b),
where Pg is an integral function of z — b, converging for sufficiently small
values of \ z — b\.'
For the vicinity of /3', we have similarly
z \z
then
and therefore
f=^" }=«.©.
:f,^}=--I
where Q2 is finite for ^^ = co .
In the vicinity of the angular point F, having a finite point on the axis of
X for its homologue, we have
W-T = e''(-+«) (z - cY R{z- c),
and, proceeding as before, we find that
r
+ R^{z- c),
Tf, ^} = *^^~^'^ ■ ^'
+
{z — c)~ z — c
where G^ depends on the coefficients in the series R{z — c).
But if the angular point F have the point at infinity on the axis of x for
its homologue, we have
F-F = e*"(-+^)-Tf-
then, proceeding as before, we find that
\W,z]=^
ia-/^^) , 1
-^^U'
where T^i-] is finite when z ^ 00
682 CURVILINEAR POLYGON [271.
Lastly, for a point W in the interior having its homologue at 2^ = z', we
have
W- W' = {z-z)S{z-z'),
and then [W,z]= S^ {z - z).
Hence [ W, z], considered as a function of z, has the following properties : —
(i) It is an analytical function of z, real for all real vaTues of the
argument z ; and if ic = 00 do not correspond to an angular
point of the polygon, then for very large values of z
where Q^ is finite when z = 00 .
(ii) It has a finite number of accidental singularities, all of them
isolated points on the axis of x : and at all other points on one
side of the plane it is uniform, finite, and continuous, having
(except at the accidental singularities) real continuous values
for real continuous values of its argument. Its form near the
singularities, and its form for infinitely large values of z, it
^ = 00 be the homologue of an angular point, are given above.
Hence {IT, ^r| can be continued across the axis of x, conjugate values of
[W, z] corresponding to conjugate values of z: and thus its properties make
it a rational meromorphic function of z.
Two cases have to be considered.
First, let the angular points of the polygon have their homologues at
finite distances from the ^-origin, say, at a, 6, . . . , ^ : and let air, ^ir, ..., Xtt be
the internal angles of the polygon at the vertices. Then
w ^|_V_^^_iV
2 —
z — a " (z — of
has no infinity in the plane ; it is a uniform analytical function of z, and
must therefore be a constant, which, by the value at z= ao , is seen to be
zero. Hence
[W,z]=X-^+^X^—^^ = 2J{z\
^ ' z — a {z -ay
the summation being for the homologues of all the angular points of the
polygon. But when z is very large, we have, in this case
F..) = ^ft(l-),
271.] REPRESENTED ON A CIRCLE 683
SO that, expanding 2J{z) in powers of - and comparing with the latter form,
we have, on equating coefficients of z'~'^, z~^, z~^,
= 2^0^' + 2a (1 - a^),
relations among the constants of the problem.
Secondly, let one angular point, say a, of the polygon have its homologue
on the axis of x at infinity, and let oltt be the internal angle at a : and let the
homologues of the others be h, ..., k, I, the internal angles of the polygon
being /Stt, . . . , kit, Xtt. Then the function
has no infinity in the plane : it is a uniform analytical function of z, and
must therefore be a constant, say M; thus
' ^ z — b{z — by
But, when z is very large, we have
tr,.) = ^
z \z
Z"
' because a; = oo is the homologue of the vertex a of the polygon, the angle
■ there being cctt ; also, T (-) is finite when z = ao . Hence, expanding in
powers of - and comparing coefficients, we have
25o = 0,
so that lW,z] = ^^ + it^—^.= 2I{z),
z — b {z — bf
where the summation is for the homologues of all the angular points other
than a, and the constants are subject to the two conditions
%B,b=^{i-o?)-\x{i-n
The form of the function { W, z] is thus obtained for the two cases, the
latter being somewhat more simple than the former : and the exact expansion
of W in the vicinity of a singular point can be obtained with coefficients
expressed in terms of the constants.
684 CEESCENT [2T2.
272. In either case the equation which determines W is of the third
order; but the determination can be simplified by using a well-known
property of linear differential equations*. If i/i and 3/2 be two solutions of
the equation
the quotient of which is equal to the quotient of two solutions of
dx-
dP
where I=Q--j P-, being the invariant of the equation for linear trans-
formation of the dependent variable, and where Y/y = e^^'^^, then the equation
satisfied by s, = y-ily-,, is
[s, x] = 21.
Hence for the present case, if we can determine two independent solutions
Z^ and Z2 of the equation
drZ
dz^
+ ZJ{z) = ,
for the first case, or two independent solutions of the equation
for the second case, then
AZ, + BZ,
CZ, + DZ,
is the general solution of the equation
{W,z] = ^J{z)oy2I{z\
and therefore is the function by which the curvilinear w-polygon is conform-
ally represented on the ^-half-plane.
273. As a first example, consider the w-area between two circular arcs
which cut at an angle Xtt. The ^r-origin can be conveniently taken as the
homologue of one of the angular points, and the ^-point at infinity along the
axis of X as the homologue of the other. Then we have
[W,z]= — + ^^^—^ — ,
' z z^
provided ^ = 0, vl . = |(1 -\-) -i(l ->^'X
both of which conditions are satisfied by J. = ; and so
'■ See my Treatise on Differential Equations, §§ 59 — 62.
273.]
CURVILINEAR TRIANGLE
685
The linear differential equation is
so that Z, = z^-^^^'^\ Z, = ^i(i-^>;
and therefore the general solution for Tl^ is
cz^ + d'
The (three) arbitrary constants can be determined by making z = and
z = QC correspond to the angular points of the crescent, and the direction of
the line z — z^ (which is the axis of x) correspond to one of the circles, the
other of the circles being then determinate.
If the w-circles intersect in — i (the homologue of the 5-origin) and + i
(the homologue of « = oo ), and if the centre of one of the circles be at the
point (cot a, 0), then the relation is
. z^ — ce~"''
w = ^ — -,
^^ + ce~°-^
where c is an arbitrary constant, equivalent to the possible constant magnifi-
cation of the 5^-plane without affecting the conformal representation : it can
be determined by fixing homologous points on the contour of the crescent.
More generally, if the w-circles intersect in w-^ and Wg) respectively homo-
logous to ^ = and z= cc , then
is the form of the relation.
Evidently a segment of a circle is a special case.
274. Next, consider a triangle in the w-plane formed by three circular
arcs and let the internal angles be Xtt, fiir, vtt. The homo-
logue of one of the angular points, say of that at yu,7r, can be
taken at z—co; of one, say of that at Xtt, at the 2^-origin ;
and of the other, say of that at vir, at a point z=l: all on
the axis of cc. Then we have
B C
z z — \
. 1 - \2
+ h 7r-+^
''{z-iy
where the constants B and G are subject to the relations
5+0=0,
5 . + (7 . 1 = i (1 - /^^) - i (1 - X^) - i (1 - v%
Fig. 97.
SO that
and therefore
-B=C=^{X^-fi? + v''-l\
iz-^y
+
^ V - yU,^ -I- V^ - 1
^ z{z-\)
686
CON FORMAL REPRESENTATION
[274.
But I{z) is the invariant of the differential equation of the hypergeometric
series *
< 1 we shall obtain merely general values of a, /3, 7 ; hence
the transforming function will be obtained as a quotient of two particular
solutions of the equation of the series. Now according to the magnitude of
\z\, these solutions, which are in the form of infinite series, change: and thus
we have w equal to an analytical function of z, which has different branches
in different parts of the plane.
The distribution of the values ^^ =^ 0, 1, 00 as the homologues of the three
angular points was an arbitrary selection of one among six possible arrange-
ments, which change into one another by the following scheme : —
1-z
1
1
1-z
z
2-1
z-\
z
1
00
1
00
1
1
00
00
00
00
1 1 1
The quantities in the first row are the homographic substitutions, conserving
the positive half-plane and interchanging the arrangements.
These substitutions are the functions of z subsidiary to the derivation of
Kummer's set of 24 particular solutions of the equation of the hypergeometric
series.
Ex. Take the case when two of the angles of the triangle are right, say v = h, ^■ = ^.
X = ~ . Then, when n is finite t, a transforming relation is
l + (i-2)*'
and, when n is infinite, a transforming relation is
1 l-(l-3)*
«<^ = log i --
l+(l-2)^
* Treatise on Differential Equations, § 116.
t ib., § 131.
and
274.] OF CURVILINEAR TRIANGLE 687
obtainable either as a limiting form of the above, or by means of the solutions F{a, ^, y, z)
and i^(a, /3, a + /3 - 7+ 1, 1 -«) of the differential equation of the hypergeometric series. In
the respective cases the general relations,' establishing the conformal representation, are
aw + by _!-{!- z)^
^«' + ^/ ~l+(l-2)4'
aw + b , l-n-z)i
, = log ^ — .
CW + d ^l + (i._^)4
The three circles, arcs of which form the triangle, divide the whole of the
w-plane into eight triangles which can be arranged
in four pairs, each pair having angles of the same .
magnitude. Thus D'
D, D' have angles Xir, /xtt, vtt. ^^
A, A' Xtt, (l-/x)7r, (l-z/)7r, / \ /'"A ^
B, B' i\ — X) TT , IXTT , {1 — v) IT ,
A
M'
and C, C (1 — X)7r, (1 — yu,) vr, z^tt ; \ \C/ B
and when any one of the triangles is given, it
determines the remaining seven. It is convenient
to choose the particular pair which has the sum of its angles not greater
than the sum for any of the others. A triangle of the selected pair is called
the reduced triangle*; let it be the triangle D. Let Sir be this smallest
sum, so that 8=\ + fx-\-v\ as the sum of the angles in each of the other
pairs of triangles is equal to, or greater than, Sir, we have
4*S^7r < sums for pairs A, B, C, J)
< Gtt,
so that ^ < I , that is, A, + //, 4- 1- < f . (The only case of exception arises when
all the four sums are equal to one another ; and then A, = ^u, = z/ = ^.)
We have already, in part, considered the case in which X + fj, + v = 1.
For, when this equation holds, inversion with the other point having Xir for
its angle as centre of inversion, changes f D into a triangle bounded by
straight lines and having Xir, /jltt, vtt as its angles; and therefore, in that
case, the problem is merely a special instance of the representation of a
w-rectilinear polygon on the ^•-half-plane.
But there is a very important difference between the cases for which
\ + [x-\-v < 1 and those for which \ + iju + v >1: in the former, the ortho-
gonal circle (having its centre at the radical centre of the three circles) is real,
and in the latter it is imaginary. The cases must be treated separately, ,^
* Schwarz, Ges. Werke, t. ii, p. 236.
t The figure in the text does not apply to this case, because, as may easily be proved, the
three circles must meet in a point.
688 FUNCTIONAL RELATION [275.
275. First, we take \ + //, + v < 1. Then of the two triangles, which
have the same angles, one lies entirely within the orthogonal circle and the
other entirely without it ; and each is the inverse of the other with regard to
the orthogonal circle *. Let inversion with regard to the angular point Xtt in
A take place : then the new triangle is bounded by two straight lines cutting
at an angle Xtt and by a circular arc cutting them at
angles /xtt and vir respectively, the convex side of the
arc being turned towards the straight angle. The
new orthogonal circle is the inverse of the old and its
centre is A, the angular point at Xir ; its radius is the
tangent from A to the arc GB, and therefore it com-
pletely includes the triangle ABC.
The homologue of A is, as before, taken to be the ^r-origin 0, that of C to
be the point z = 1, say c, and that of 5 to be ^ = co on the axis of x, say b for
+ 00 and 6' for — 00 .
Suppose that we have a representation of the triangle on the positive
half-plane of z. The function [w, z] can be continued across the axis of x
into a negative half-plane, if the passage be over a part of that axis, where
the function is real and continuous, that is, if the passage be over Oc, or over
c&, or over h'O ; and therefore w is defined for the whole plane by [w, z] = 2/ (z),
its branch -points being 0, a, b. Any branch on the other side, say w^, will
give, on the negative half-plane, a representation of a triangle having the
same angles, bounded by circular arcs orthogonal to the same circle, and
having 0, c, b for the homologues of its angular points. Thus if the con-
tinuation be over cb, the new w-triangle has CB common with the old, and
the angular point J.' lies beyond CB from A.
To obtain the new triangle A 'CB geometrically, it is sufficient to invert
the triangle AGB, with regard to the centre of the circular arc CB. This
inversion leaves CB unaltered; it gives a circular arc CJ.' instead of CA
and a circular arc BA' instead of BA: the angles of A'CB are the same as
those of A CB. Since the orthogonal circle of A CB cuts CB at right angles
and CB is inverted into itself, the orthogonal circle is inverted into itself;
therefore the triangle A'CB has the same orthogonal circle as the triangle
ACB.
The branch lu^ , by passing back across the axis round a branch-point into
the positive half-plane, leads to a new branch lUo, which gives in that half-plane
a representation of a triangle, again having the angles Xtt, /mtt, vtt and having
0, c, b for the homologues of its angular points. Thus if 'the passage be
over Oc, the new w-triangle has A'C common with A'CB and the angular
point B" lies on the side of CA' remote from B : but if the passage be
* For the general properties of such systems of circles, see Lachlan, Quart. Journ. Math.,
vol. xxi, (1886), pp. 1—59.
275.] FOR CURVILINEAR TRIANGLE 689
over cb, then we merely revert to the original triangle CAB. The new
triangle has, as before, the same orthogonal circle as A'GB.
Proceeding in this way by alternate passages from one side of the
axis of X to the other, we obtain each time a new w-triangle, having one side
common with the preceding triangle and obtained by inversion with respect
to the centre of that common side : and for each triangle we obtain a new
branch of the function w, the branch-points being 0, 1, oo . If, by means of
sections such as Hermite's (§ 103), we exclude all the axis of x except the part
between two branch-points, the function is uniform over the whole plane thus
bounded.
All these triangles lie within the orthogonal circle, and they gradually
approach its circumference : but as the centres of inversion always turn that
circle into itself, while the sides of the triangle are orthogonal to it, they do
not actually reach the circumference. The orthogonal circle forms a natural
limit (§81) to the part of the w-plane thus obtained.
Ex. Shew that all the inversions, necessary to obtain the complete system of triangles,
can be obtained by combinations of inversions in the three circles of the original triangle.
(Burnside.)
Each of the triangles, thus formed in successive alternation, gives a
w-region conformally represented on one half or on the other of the 2^- plane.
If, then, the original triangle be combined with the first triangle that is
conformally represented on the negative half-plane, every other similar
combination may be regarded as a symmetrical repetition of that initial
combination : each of them can be conformally represented* upon the whole
of the ^-plane, with appropriate barriers along the axis of x.
The number of the triangles is infinite, and with each of them a branch
of the function w is associated: hence the integral relation - between w
and z which is equivalent to the differential relation [w, z] = 2I (z), when
\ + /J, + v kI, is transcendental in w.
In the construction of the successive triangles, the successive sides passing
through any point, such as C, make the same angle each with its predecessor :
and therefore the repetition of the operation will give rise to a number of
triangles at G eaqh having the same angle A-tt.
If X, be incommensurable, then no finite number of operations will lead to
the initial triangle : each operation gives a new position for the homologous
side and ultimately the w-plane in this vicinity is covered an infinite number
of times, that is, we can regard the w-surface as made up of an infinite
number of connected sheets.
If A, be commensurable, let it be equal to l/l', where I and I' are finite
integers, prime to each other. When I is odd, 21' triangles will fill up the
w-space immediately round C, and the {21'+ l)th triangle is the same as the
F. F. 44
690 SPECIAL [275.
first : but the space has been covered I times since ^I'Xir = 2^7r, that is, in the
vicinity of C we can regard the w- surface as made up of I connected sheets.
When I is even (and therefore V odd), V triangles will fill up the space round
G completely, but the (Z'+ l)th triangle is not the same as the first: it is
necessary to fill up the space round G again, and the (2Z'+l)th triangle is
the same as at first ; the space has then been covered I times, so that again
the w-surface can be regarded as made up of I connected sheets. The
simplest case is evidently that in which X is the reciprocal of an integer, so
that l = \\ and the w-surface must be regarded as single-sheeted.
Similar considerations arise according to the values of /u, and of v.
If then either A,, fx, or v be incommensurable, the number of t«-sheets is
unlimited, that is, ^ as a function of w has an infinite number of values, or the
equation between z and w is transcendental in z. Hence, when X. + yu, + y < 1
and either \ or /j, or v is incommensurable, the integral relation between w and
z, luhich is equivalent to the differential relation [w, z] =21 (z), is transcend-
ental both in tv and in.z.
If all the quantities X, (x, v he commensurable, the simplest case of all
arises when they are the reciprocals of integers*. Then ^ is a uniform
transcendental function of w, satisfying the equation
{w,z] = 2I{z);
or, making z the dependent and w the independent variable, we have the
result : —
A function z that satisfies the equation
1
/V 1 " n^ . P' m- n
d^z dz ^ fd^zy
dw^ dw ^ \dw^J
1 1 1 1 111/
fi'-Y
2 ^2 +h(^2_iy+l- ^(^_i) ]\dwj'
111
where I, m, n are integers, such that -j+~-\--yi dcji, K' =\^ [I -{I - z)sin^ (f)]'^ d(f>,
.' JO
* The figure for the example v = \, /J. = i, X = i is given by Schwarz, Ges. Werke, t. ii, p. 240;
and the figure for the example v = ^, ,u.=i, X = 7 is given in Klein-Fricke (p. 370); both of course
satisfying the conditions 'K + fi + v 1 from the pair D and D',
— \ + /u, + vS are the middle points of the
edges : all projected from the centre of the sphere.
The shaded triangles (the visible twelve being one half of the aggregate)
correspond to one half of the ^-plane ; the unshaded triangles correspond to
the other half of the 5-plane.
Each of the angles at is ^tt : each of the angles at C is ^tt : each of
the angles at 8 \b ^ir; and it may be noted that the triangles GOG are the
triangles in the tetrahedral division of the spherical surface, the point
696 CONSTRUCTION [2*77.
in the present triangle COC being the point ^ in a triangle STF, and the
two points G being the points F and T in the former figure (fig. 102).
0|6
The solid is the icosahedron or the dodecahedron. These two solids can
be placed so as to have the same planes of symmetry, by making the centres
of the twenty faces of the icosahedron the vertices of the dodecahedron. In
the figure (fig. 104) the vertices of the icosahedron are the points I: those
of the dodecahedron are the points D : and the middle points of the edges
. are the points >S^. The shaded triangles (the visible thirty, six in each lune
through the vertex of the icosahedron, being one half of their aggregate)
correspond to one half of the ^-plane : the unshaded triangles, equal in
number and similarly distributed, correspond to the other half of the ^■-plane.
The angles at the vertices I of the icosahedron are ^tt ; those at the vertices
D of the dodecahedron are ^v ; and those at the middle points S of the edges
(the same for both solids) are -^tt.
278. Having obtained the division of the surface, we now proceed to
determine the functions which establish the conformal representation.
In all these cases, ^ is a rational function of w : therefore when we
know the zeros and the infinities of ^ as a function of w, each in its proper
degree, we have the function determined save as to a constant factor. This
factor can be determined from the value of tu when z = l.
278.] OF TRANSFORMING RELATIONS 697
The variable tu belongs to the stereographic projection of the point of
the spherical surface on the equatorial plane, the south pole being the pole
of projection. If X, Y, Z be the coordinates of the point on the spherical
surface, the radius being unity, then
X + iY
^ = TTX-
For a point in longitude I and latitude ^tt — 8, we have X = cos ^ sin S,
Y = sin 7 sin 8, Z = cos 8 : so that, if preferable, another form for w is
IV = e»'^ tan ^8.
In our preceding investigation, the angle at Xir was made to correspond
with z = 0, that at vir with 2=1, that at /j^tt with z= cc .
Case I. We take X,= -, ix = ^, v = I.
^77" 47r
For the angular points /i7r we have 8 = \tt\ 1 = 0, — , — , . . . , each pomt
belonging to two triangles of the same set, that is, triangles represented on
the same half of the plane : thus the various ty-points in the plane are
2712 ^
for r = 0, 1, ..,,« — 1, each occurring twice. Hence z = cc , when the function
w— 1 '^
n {w-e"" ''y
r=0
vanishes, that is, s = oo , when (w" — 1)'-* vanishes.
For the angular points vir, we have S = -^7r; 1=—, — , — , ..., each
point belonging to two triangles of the same set : thus the various w-points
in the plane are
^\2r + l)
for r = 0, 1, ..., n — 1, each occurring twice. Hence z = l, when the function
H [w-e"" Y
vanishes, that is, z=l, when {w''^ + 1)^ vanishes.
Now ^ is a uniform function of lu : hence v^e can take
where ^ is a constant, easily seen to be unity : because, when u> =
(corresponding to the common vertex Xtt at the North pole) and when
698 TETRAHEDRAL FUNCTION [278.
w = 00 (corresponding to the common vertex Xtt at the South pole), z vanishes,
as required. The relation is often expressed in the equivalent form
z -.2-1 -.1 = - 4>w'' : - (w" + 1)^ : (w" - l)^
which gives the conformation on the half ^^-plane of a w-triangle bounded by
circular arcs, the angles being -, ^tt, ^tt. The simplest case is that in
77"
which the triangle is a sector of a circle with an angle - at the centre.
The preceding relation is a solution of the equation
I1-- --1I
(z-iyz(z-i)]-
If we choose X = ^, /^ = ^, v = -; so that z = when (w" + l)^ vanishes,
2: = 00 when {w'^—Vf vanishes, and z=l when w"^ vanishes, the relation
establishing the conformal representation is
z : z-1 : 1 = (w'» + 1)^ : 4^^* : {w-^ - If.
This relation is a solution of the equation
[lU,
+ T TT„ +
z- (z — If z {z — l)j
Case II. We take A, = I ; so that z = must give the points S, each of
them twice, since there are two triangles of the same set at ;S : yu- = ^ (and
these are taken at T), so that z = co must give the points T, each of them
thrice : and v = ^ (and these are taken at F), so that ^ = 1 must give the
points F, each of them thrice.
Taking the plane of the paper as the meridian from which longitudes are
measured, the coordinates of the four w-points in the plane, corresponding to
T by stereographic projection, are
V2 _V2 .x/2 _ .V2
V3 V3 'V3 \/3
V3 V3 '^V3 "^V3
say w^, W2, Ws, Wi. Then z=oo gives each of these points thrice: that is,
z= , when {(w — Wi) ... (w — iUi)Y vanishes, or 2^ = 00 , when
(iv' - 2w- V3 - 1)=^
vanishes.
278.] OCTAHEDRAL FUNCTION 699
The coordinates of the four points corresponding to F, are
V2
V2 . V2 . \/2
\/3 ^3 V3
1^73'
1 ' 1 ' 1
^V3 V3 V3
Hence z—1, when
(w^ + 2W^ \/3 - 1)'
vanishes.
(2j.+i)-:if
The coordinates of the six points corresponding to S are 0, e ^ (for
?' = 0, 1, 2, 3) and X) : hence z = Q, when
vanishes.
Moreover, 2; is a uniform function of w : and therefore
_ ( w* + 2w^ V3 -^ 1)=^
^~(w^-2tt;V3-l)''
the constant multiplier on the right-hand side being determined as unity by
the relation between the points S and the value z = 0.
The relation is often expressed in the equivalent form
z :z-\ : l = 12V3w2(w*+l)' : K + 2wW3-l)' : -{w'-2iu- ^J^-l)\
It gives the conformation on the 2;-half-plane of a triangle in the w-plane,
bounded by circular arcs, the angles of the triangle
being ^-tt, Jtt, ^tt.
The simplest case is that of a portion cut out ,,--'
of a sector of a circle of central angle 30°, by the ,--''
arc and two lines at right angles to one another ^
symmetrical with respect to the arc. ^'
It has been assumed that the plane of the paper is the meridian.
Another convenient meridian to take is one which passes through a point
8 on the equator : in that case, the preceding analysis applies if a rotation
through an angle ^tt be made. The effect of this rotation is to give the new
variable W for any point in the form
in
so that 'uf = — iW^. The relation then takes the form
'z :z-l : 1
= 12 V^ W\W'-iy:(W' + 2W"^^+iy : -(W'-2W'^'^ + iy;
but there is no essential difference between the two relations.
700 OCTAHEDRAL FUNCTION [278.
The lines by which the w-plane is divided into triangles, each conformally
represented on one or other half of the ^-plane, are determined by z = Zo,
that is, by
The figure is the stereographic projection of the division of the sphere, and
it can be obtained as in § 257 (Ex. 20, Ex. 23).
Case IV. We take X = i, so that z=0 must give the eight points C :
each is given three times, because at C there are three triangles of the same
set : we take v = i, so that z = l must give the six points 0, each four times :
and /i = I, so that z = cc must give the twelve points S, each of them twice.
We take the plane of the paper as the meridian. The points are 0, 1,
i, — 1, — ^, 00 ; each four times. Hence z=l, when the function
vanishes.
+ 1 + i
The points C are the eight points ~ ~ : the product of the eight
i V " ~ 1
corresponding factors is
w^ + 14^4 + 1 :
and each occurs thrice, so that z = 0, when the function
vanishes.
The points S are (i) the four points — =^ — =- in the plane of the paper,
giving a corresponding product
w* - Qw^ + 1 :
+ i .
(ii) the four points ~ — - in the meridian plane, perpendicular to the
plane of the paper, giving a corresponding product
lu^ + Quj- + 1: ■
and (iii) the four points e* , (for r = Q, 1, 2, 3), in the equator, giving a
corresponding product
w*+ 1.
Each of these points occurs twice : and therefore .2 = 00, when the function
that is, when the function
(lu'-' - ssw' - ssw' + ly-
vanishes.
278.]
Hence
z =
OCTAHEDRAL FUNCTION
(w^ + 14w'' +1)='
701
(wi2-33w«-33w^+l)2'
the constant multiplier being determined as unity, by taking account of the
value unity for z : and
_ _ 108w^ {w^ - iV
(w^^ - 33 w« - ;i3tt;* + 1)'-' ■
The relation can be expressed in the equivalent form
z:z-\:\ = {'ufi-\- 14w* + 1)^ : lOSit;^ {w^ - Vf : (w^^ - 33w» - 33w* + Yf.
It gives the conformation on half of the ^-plane of a w-triangle bounded by
circular arcs and having its angles equal to \'k, \ir, \'k respectively.
The lines, by which the ?<;-plane is divided into the triangles, are given
by z = z^, that is, by
w*{w*- ly
Wo' {wo* -ly
Fig. 106.
The division is indicated in fig. 106, being the stereographic projection of
the divided spherical surface of fig. 103, with respect to the south pole,
taken to be diametrically opposite to the central point 0.
Case VI. We take \ = |^, so that z = must give the twenty points D,
each of them thrice; v = ^, so that z = l must give the twelve points I, each
702 ICOSAHEDRAL [278.
of them five times ; and yu. = i , so that ^ = oo must give the thirty points S,
each of them twice.
Let an edge of the icosahedron subtend an angle 6 at the centre of the
sphere: then its length is 2rsm^0. Also, five edges are the sides of a
pentagon inscribed in a small circle, distant d from a summit : hence the
radius of this circle is r sin 6 and the length of the edge is 2rsin ^siniTr,
so that
2 sin ^6 = 2 sin sin ^tt,
whence tani6' = i (\/5 - 1), cot i^ = i (V5 + 1).
2771
Let a denote e'^^. Then the value of w corresponding to the north pole /
is ; the values of tu for the projections on the equatorial plane of the five
points / nearest the north pole are
tan ^6, aHan^^, a" tan ^^, aHan^^, ocHan^^:
the values of w for the projections on the equatorial plane of the five points /
nearest the south pole are
a cot ^ 6, a^ cot h 6, o? cot \ d, (x! cot \ 6, o? cot i ^ :
and for projection of the south pole the value of w is infinity. The product
of the corresponding factors is
4 4
w . n (w-a2'-tani6') . 11 (w - a^'^+i cot i ^) . 1
= w{w^- t&iv' ^d) (w' + cot^ -|- 6)
= w {v:^° + Uw' - I),
after substitution. Each point / occurs five times ; and therefore z = l, when
the function
vanishes.
The points D lie by fives on four small circles with the diameter through
the north pole and the south pole for axis. The polar distance of the small
circle nearest the north pole is tan S = 8 — V5, and of the circle next to it is
tan 8' = 3 + \/5, so that
V1.5-6V5-1 , V15 + 6V5-1
tan .^6 = rp , tan 16 = ^-- — .
The function corresponding to the projections of the five points nearest the
north pole is
w^ + tan' ^ S,
and to the projections of the five nearest the south pole is
w'^ — cot^ ^S;
278.]
FUNCTION
703
while, for the projections of the other two sets of five, the products are
w^ + tan^iS'
and w'-cot'^8'
respectively. Each occurs thrice. Hence z = 0, when the function
{{w' + tan^S) (w5 _ cot^ |g) (^s + tan^S') (W - cot^ ^S')Y,
vanishes, that is, when (w^-' - 228w'' + 4^Mtv^o ^ 22Siv' + ly, which is the
reduced form of the preceding product, vanishes.
Fig. 107
The points S lie by tens on the equator, by fives on four small circles
having the polar axis for their axis. Proceeding in the same way with the
products for their projections, it is found that ^= oo , when the function
[w^' + 1 + 522^5 (,^^20 _ 1) _ lOOOotw^o (w'° + 1)Y
vanishes.
704 THE FIFTEEN RATIONAL [278.
(w^o - 228^15 ^ 494^1° + 228w' + 1 )^
Hence z = |,^3o + i + 522m;5 (w^" - 1) - 10005w>» (w" + 1)1' '
the constant factor being found to be unity, through the value of 1 — ^
which IS l-z = 1^30 ^_ 1 + 522^-5 (lu^-o _ 1) _ 10005m;" (w^" + 1)]^ '
These relations give the conformal representation on half of the ^-plane of a
wrtriangle, bounded by circular arcs and having angles Itt, ^tt, ^tt.
The lines, by which the w-plane is divided into the triangles, are given
by z = 2o, that is, by
(w-^« - 228^15 + 494m;1o + 228w^ + 1)^ _ (wp^" - 228<^ + 494wo^« + 228wo^ + 1 )^
w' {w'' + iiiu' - ly ~ Wo'{wo'' + uwo'-iy
The division is indicated in figure 107, which is the stereographic projection* of
the divided spherical surface of figure 104, with I^^ as the pole of projection.
279. The preceding are all the cases, in which simultaneously ^ is a
uniform function of w, and w is an algebraical function of z : they arise when
the surface of the sphere has been completely covered once with the two sets
of triangles corresponding to the upper half and the lower half of the ^-plane.
But an inspection of the figures at once shews that they are not the
only cases to be considered, if the surface of the sphere may be covered
more than once.
In the configuration arising through the double-pyramid, the surface of
the sphere will be covered completely and exactly m times, if the angles at
the poles be Imirjn, where m is prime to n. The corresponding relation
between w and z is obtained from the simpler form by changing n into nj.m.
In the tetrahedral configuration (fig. ]02) the surface of the sphere will
be exactly and completely covered twice by triangles FFT (or by triangles
TTF, it being evident that these give substantially the same division of the
surface). The relation between w and z will then be of the same degree,
12, as before in w, for the number of different triangles in the two ^y-sheets
is still twelve of each kind : because there are two w-sheets corresponding
to the single ^^-plane, that relation will be of the second degree in z. The
values of the angles are determined by
(III.) \,^.,v = l,\,\.
* lu regard to all the configurations thus obtained as stereographic projections of a spherical
surface, divided by the planes of sjmmetiy of a regular solid, Mobius's " Theorie der symme-
trischen Figuren" [Ges. Werke, t. ii, especially pp. 642 — 699), may be consulted with advantage ;
and Klein-Fricke, Elliptische Modulfunctionen, vol. i, pp. 102 — 106.
279.] TRANSFORMING FUNCTIONS 705
Again, in the octahedral configuration, the surface of the sphere will
be exactly and completely covered twice by triangles OCO! The relation
between w and z will be of degree 24 in tu and degree 2 in ^ : and the values
of the angles are determined by
(V.) X,f.,v=ll,l
Similarly, a number of cases are obtainable from the icosahedral configu-
ration, in the following forms :
(VII.) \ /Jb, V = ^, i, J with triangles such as IiD^D.^;
(VIII.) \,f.,v = hh^ A/J.;
(IX.) \,x,v = ^,l^ SJJ,;
(X.) \f.,v = l,hi DJJ,;
(XL) x,/., z. = |, f,| IJJ,;
(XII) \ /z, z. = f , 1, 4 /lAAs;
(XIII.) \,f,,r; = A,i,± IJJ,,;
(XIV.) \,fi,v = ^,ii 7AA;
(XV.) \, /., j. = f,i, 1 IJ,D,.
Other cases appear to arise : but they can be included in the foregoing,
by taking that supplemental triangle which has the smallest area. Thus,
apparently, I^DJ^f^ would be a suitable triangle, with ^, /"-, ?^ = f , f , ^: it is
replaced by IioD-zo^iot an example of case (X.) above.
These, with the preceding cases numbered* (I.), (II.), (IV.), (VI.), form
the complete set of distinct ways of appropriate division of the surface
of the sphere.
It is not proposed to consider these cases here : full discussion will be
found in the references already given. The nature, however, of the relation,
which is always of the form
f(z) = F(w),
where / and F are rational functions, may be obtained for any particular
case without difficulty. Thus, for (III.), we have
[w, z\ =
when
_ 1
2
"1—1 1_1 l_l-4-i — 1'
z^ '^ {1-zf z{z-\) ]'
z■.l-z■.l = -li^/^wHw'^-lf: (w* + 2w;V3- 1)' : (w^ - 2t^V3 - 1)^
Again, if
z: 1 - z : 1 = (Z + iy : - 4^Z : {Z -ly,
* These numbers are the numbers originally assigned by Schwarz, Ges. Werke, t. ii, p. 246,
and used by Cayley, Camb. Phil. Trans., vol. xiii, pp. 14, 15.
F F. 45
706 POLYHEDRAL FUNCTIONS [279.
a special case of § 278, I., by taking n = 1, then
'dzY
Hence
iQ(z + iy
(i-i)^. i(i-i) "
_ 1
2
(Z-iy iii-^y z{z-i)_
"1—1 1—4^ l — i-f-l— 1
9 I ■'■ 9 I 9 9^9 -*
Z' ^{Z-iy^ z{Z-i) J'
so that X. = -3-, v=^, yU' = j. Hence the relation
{Z+iy:-4>Z:{Z-iy
= - 12 V3 w^ (w* + 1)^- : {w' + 2tv'~ ^S -iy:(w'- Iw" V3 - 1)'
gives the conformation of triangles bounded by circular arcs and having
angles Jtt, ivr, f tt.
The foregoing are the only cases, for A, + yu, + v > 1, in which the integral
relation between w and z is rational both in w and in z.
In all other cases in which X, /x, v are commensurable, this integral
relation is rational in z and transcendental in iv.
It is to be noticed, in anticipation of Chapter XXII., that, since every
triangle in any of the divisions of the spherical surface, or of the plane,
can be transformed into another triangle, the functions which occur in
these integral relations are functions characterised by a group of substi-
tutions. When the functions are rational, the groups are finite, and
the functions are then the polyhedral functions : when the functions are
transcendental, the groups are infinite, and the functions are then of the
general automorphic type.
The case in which \+ /j,+ v = 1 has already been considered : the spherical
representation is no longer effective, for the radius of the sphere becomes
infinite and the triangle is a plane rectilinear triangle. The equation may
still be used in the form
{w,z} = 2I(z),
with the condition \-\- ibi + v = l. A special solution of the equation is then
given by
-r- = z^-' (1 - zY-\
dz
leading to the result of § 268, the homologue of the angular point fjuir being
at ^ = 00 .
280. It is often possible by the preceding methods to obtain a relation
between complex variables that will represent a given curve in one plane on
280.] ALGEBRAIC ISOTHERMAL CURVES 707
an assigned curve in the other : there is no indication of the character of the
relation for an arbitrary curve or a family of curves. But in one case, at any
rate, it is possible to give an indication of the limitations on the functional
form of the relation.
Let there be a family of plane algebraical curves, determined as potential
curves by a variable parameter*: and let their equation be
F {x, y, u) = 0,
where u is the variable parameter, which, when it is expressed in terms of x
and y by means of the equation, satisfies the potential-equation
Since u is a potential, it is the real part of a function w oi x-\- iy : and the
lines u = constant are parallel straight lines in the w-plane. It therefore
appears that the functional relation between w and z must represent the
w-plane conformably on the ^-plane, so that the series of parallel lines in the
one plane is represented by a family of algebraical curves in the other : let
the relation, which effects this transformation, be
X {z, w) = 0.
Let the algebraical curve, which corresponds to some particular value of u,
say u = 0, be
F{x,y,0)=f{x,y)^0,
which in general is not a straight line. Let a new complex ^ be determined
by the equation
f[l
=
this equation is algebraical, and therefore ^ can be regarded as a function of
lu, say t/t (w), between which and z, regarded as a function of w, say {w),
there is an algebraical equation.
Now when u — 0,z describes the curve
f{x,y)=0:
hence at least one branch of the function ^, defined by
* Such curves are often called isothermal, after Lame. The discussion of the possible func-
tional relations, that lead to algebraical isothermal curves, is due to Schwarz, Ges. Werke, t. ii,
pp. 260 — 268: see also Hans Meyer, " Ueber die von geraden Linien und von Kegelschnitten
gebildeten Schaaren von Isothermen ; so wis iiber einige von s-peciellen Curven dritter Ordnung
gebildete Schaaren von Isothermen," (a Gottingen dissertation, Ziivich, Zlircher and Furrer,
1879); Cayley, Quart. Journ. Math., vol. xxv, (1891), pp. 203—226, Coll. Math. Papers, vol. xiii,
pp. 170 — 191 ; and the memoir by Von der Miihll, cited p. 611.
45—2
708 FAMILIES OF [280.
can be taken as equal to x when u = 0, that is, there is one branch of the
function ^ which is purely real when w is purely imaginary.
The curves in the ^-plane are algebraical : when this plane is conformally
represented on the ^-plane by the foregoing branch, which is an algebraical
function of z, the new curves in the ^-plane are algebraical curves, also
determined as potential curves by the variable parameter u. And the ^-curve
corresponding to i< = is (the whole or a part of) the axis of real quantities.
In order that the conformal representation may be effected by the functions,
they must allow of continuous variation : hence lines on opposite sides of
u = correspond to lines on opposite sides of the axis of real quantities. The
functional relation between ^= | + ^»7 and w = u-\- iv is therefore such that
^ + ir} = '^jr (u + iv),
^ — irj^yjr (— u + iv).
The equation of the ^-curves, which are obtained from varying values of
u, is algebraical : and therefore, when we substitute in it for f and 77 their
values in terms of y^ {u + iv) and i/r (— w + iv), we obtain an algebraical
equation between -^ {u + iv) and i/r (— t^ + iv), the coefficients of which are
functions of u though not necessarily rational functions of lo. Let 6 = — 2u;
and let yjrz, •\/^3 denote -^{w), ylr(w + d) respectively; then the equation can
be represented in the form
g(ir„ir.,0) = 0,
rational in 1^2 and yps, but not necessarily rational in 6.
Because the functions allow continuous variation, we can expand -1/^3 in
powers of 6 : hence
,(^.,^.,.^^,,^*^, ^)=o.
When this equation, which is satisfied for all values of w and of 6, where
w and 6 are independent of one another, is arranged in powers of 6, the
coefficients of the various powers of 6 must vanish separately. The coefficient
independent of 6, when equated to zero, can only lead to an identity, for it
will obviously involve only yfr^ : any non-evanescent equation would determine
i/to as a constant. Similarly, the coefficient of every power of 0, which
involves none of the derivatives of yjr^, must vanish identically. The co-
efficient of the low^est power of 6, which does not vanish identically, involves
"^2. -y— " and constants: but, because the equation g (^jr^, ■^s, 6) = is
rational in i/tj, the second and higher derivatives of yfr^, associated with the
second and higher powers of 6 in the expansion of -v/tj, cannot enter into the
coefficient of this power of 6. Hence we have
«Vh'>'
280.] ALGEBRAIC ISOTHERMAL CURVES 709
an algebraical equation between o/to and -~ , the coefficients of which are
constants.
The coefficient of the next power of 6 will involve —^ , and so on for the
powers in succession. Instead of using the equations, obtained by making
these coefficients vanish, to deduce an algebraical equation between -v^a
and any one of its derivatives, we use A = 0. Thus for ~~^ , the equation
would be obtained by eliminating -v/r/ between the (algebraical) equations
and so for others.
Returning now to the equation
in which, as it is rational in -v/r, and yfr^, only a limited number of co-
efficients, say k, are functions of 6, we can remove these coefficients as
follows. Let k — 1 differentiations with regard to w be effected : the resulting
equations, with g = 0, are sufficient to determine these k coefficients ration-
ally in terms of -v/^a, yfra and their derivatives. But the coefficients are
functions of 6 only and do not depend upon w : hence the values obtained for
them must be the same whatever value be assigned to w. Let, then, a zero
value be assigned : yjr^ and its derivatives become constants ; yjrs becomes
ylr{6), say -v/rj, and all its derivatives become derivatives of i/tj; so that the
coefficients can be rationally expressed in terms of -x/^j and its derivatives.
When these values are substituted in g — 0, it takes the form
9i (^2, i^s,^!, fi, -^i", ■■■) = 0,
rational in each of the quantities involved. But between yjr^ and each
of its derivatives there subsists an algebraical equation with constant co-
efficients : by means of these equations, all the derivatives of yjri can be
eliminated from g^ = 0, and the final form is then an algebraical equation
involving only constant coefficients. But
^fr,=^^lr (0), ^|r,= ^Jr (w), f, = ir(w + d);
and therefore the function ^|r (w) possesses an algebraical addition-theorem.
Now yjr (w) and 4> {w) are connected by the algebraical equation
therefore ^{w) possesses an algebraical addition -theorem. But, by § 151,
710 FAMILIES OF [280.
when a function K
x^
_ 1
4:
SO that , _ 1 1. „ "*" , _^ ^.„
{q ^ + q^y (q i^-q^y
an ellipse, agreeing with the result in | 257, Ex. 7. This is obtained from
the relation
' il-w /2K . , \
k 2 = sn — sm ^ z ] ,
1+W \ TT 1
which is not included in the general forms of relation obtained in the
preceding investigation.
712*
SUEFACES OF
[280.
But the equation
yfctsnf— ^) +
[7I /2^ „
does not lead to an algebraical relation between x and y for a general (non-
zero) value of u. Neither the conditions of the earlier proposition nor its
limitations apply to this case.
The problem of determining the kinds of functional relation which will
represent a single algebraical curve in the 5-plane upon a single line of the
w-plane is wider than that which has just been discussed : it is, as yet,
unsolved.
Note on § 275 (see foot-note, p. 690).
The investigations in §§ 276-279 shew how important, in the development
of the polyhedral functions connected with the hypergeometric series, is the
surface of a sphere, that is, a surface of constant positive curvature. All the
cases in which algebraic expression can arise for \ + /bu + v > 1 are thereby
considered.
It might seem not improbable that a corresponding use of surfaces of
constant negative curvature could be made for X + fM + v < 1 ; but the whole
investigation is much more difficult, because the relation between w and z is
always transcendental in one of the two variables. The following propositions
are worthy of record.
A. All surfaces of given constant (negative) curvature are deformable
into one another, without stretching or tearing*.
Consequently, it is sufficient to take any one of
them as a surface of reference ; and the simplest,
as regards the geometrical property, appears to be
the surface of revolution formed by the rotation
of the plane tractrix — the symmetrical involute of
a catenary — about its asymptote which is the
directrix of the catenary. The equations of the
generating curve ■[- are
Xo = a sin cf), yQ = a (cos ^ + log tan ^ cf)) ;
the range of ^ for the upper part of the (dotted)
curve is tt to -|-7r, and for the lower part is J-tt to 0.
The curve is periodic analytically, so far as con-
cerns (f), with a period 27r ; but geometrically it is
* A well-known theorem, originally due to Minding, on the basis of a theorem proved by
Gauss ; see my Lectures on Differential Geometry, §§ 211, 212.
t See Darboux, Theorie generate des surfaces, (t. iii, pp. 394 sqq.), for a discussion of the
surface.
280.] CONSTANT NEGATIVE CURVATURE 713
imaginary, and therefore also the sheet of the surface is imaginary, for the
half-period ir to 2it. and for every corresponding half-period. And there is
an infinite number of sheets, real and imaginary.
The arc Sq of the tractrix from its vertex is given by
So = <-i log cosec <^.
The arc of the surface of revolution is given by
ds- = dSff' + x^ d&^
= -, {dxi-^ + de%
where u =
sin<^ '
The surface is isothermal in terms of these variables u and Q ; the range
of ^ is to Stt, and the range of u (for ^ between ^tt and 0) is from 1 to oo .
The area of the upper half of the surface is the same as that of the lower
half; each of them
= I 277 Xndsg = 2'7ra^.
Jo
For the Gaussian measure of curvature of the surface of revolution, we have
p = — a cot S^ S\ ....
But we have negative powers of S also. The definition of S° (z) is
given by
SS'>{z) = S(z),
so that So (z) = z and it is often called the identical substitution : the
definition of S~^ (z) is given by
SS-^z) = S'>{z) = z,
■ {■ -r a / \ Cl^ + b
SO that 8-1 (z) is a substitution the inverse of S ; m fact, it w = S{z)= , ^ >
then z = S-Hu^ ~ ^ And then, from S-'^z, by repetition we obtain
cw — a
s-\s-\s-\ ....
If some of all the substitutions to which a variable z is subject be
not included in 8 and its integral powers, then we have a new substitution
T and its integral powers, positive and negative. The variable is then
subject to combinations of these substitutions : and, as two general linear
substitutions are not interchangeable, that is, we do not have T{8z) = 8{Tz)
in general, therefore among the substitutions to which z is subject there
must occur all those of the form
.„8^T^SyT'...,
where a, /3, ^,8, ... are positive or negative integers.
If, again, there be other substitutions affecting z, that are not included
among the foregoing set, let such an one be U: then there are also powers
of U and combinations of 8, T, U (with integral indices) operating in any
order: and so on. The substitutions >S^, T, U, ... are called fundamental :
the sum of the moduli of a, ^S, 7, 8, ... of any substitution, compounded from
* Corns d'Algebre Superieure, t. ii. Sect, iv, (Paris, Gauthier-Villars).
f Traite des substitutions, (lb., 1870).
J Substitutionentheorie und ihre Anwendung auf die Algebra, (Leipzig, Teubner, 1882).
§ Vorlesungen uber das Ikosaeder, (ib., 1884).
II Theory of groups of finite order, (Cambridge, University Press, 2nd ed., 1911).
** Math. Ann., t. xxi, (1883), pp. 141 — 218, where references to earlier memoirs by Klein are
given.
ft Acta Math., t. i, (1882), pp. 1—62, pp. 193—294; ib., t. iii, (1883), pp. 49—92.
Xt Math. Ann., t. xx, (1882), pp. 1—44; ib., t. xxii, (1883), pp. 70—108.
§§ Amer. Jottrn. of Math., vol. xiii, (1890), pp. 59—144.
281.] GROUPS OF SUBSTITUTIONS 7l7
the fundamental substitutions, is called the index of that substitution ; and
the aggregate of all the substitutions, fundamental and composite, is the
group.
There may however be relations among the substitutions of the group,
depending on the fundamental substitutions ; they are, ultimately, relations
among the fundamental substitutions, though they are not necessarily the
simplest forms of those relations. Hence, as we may have a relation of
the form
...lS^...T^...U'...{z) = z,
the index of a composite substitution is not a determinate quantity, being
subject to additions or subtractions of integral multiples of quantities of the
form (a) + (6) + (c) + ..., there being one such quantity for every relation:
we shall assume the index to be the smallest positive integer thus obtainable.
282. There are certain classifications which may initially be associated
with such groups, in view of the fact that the arguments are the arguments
of uniform automorphic functions satisfying the equation
f{Sz)=f{z):
in this connection, the existence of such functions will be assumed until their
explicit expressions have been obtained.
Thus a group may contain only a finite number of substitutions, that is,
the fundamental substitutions may lead, by repetitions and combinations, only
to a finite number of substitutions. Hence the fundamental substitutions,
and all their combinations, are periodic in the sense of § 260, that is, they
reproduce the variables after a finite number of repetitions.
Or a group may contain an infinite number of substitutions : these may
arise either from a finite number of fundamental substitutions, or from an
infinite number. The latter class of infinite groups will not be considered
in the present connection, for a reason that will be apparent (p. 732, note)
when we come to the graphical representations. It will therefore be
assumed that the infinite groups, which occur, arise through a finite number
of fundamental substitutions.
A group may be such as to have an infinitesimal substitution, that is,
there may be a substitution — -y , which gives a point infinitesimally near
to z for every value of z. It is evident there will then be other infinitesimal
substitutions in the group ; such a group is said to be continuous. If there
be no infinitesimal substitution, then the group is said to be discontinuous,
or discrete.
But among discontinuous groups a division must be made. The definition
of group-discontinuity implies that there is no substitution, which gives an
infinitesimal displacement for every value of z: but there may be a number
718 ' DISCONTINUOUS GROUPS [282.
of special points in the plane for regions in the immediate vicinity of which
there are infinitesimal displacements. Such groups are called improperly
discontinuous in the vicinity, of such points: all other groups are called
properly discontinvous. For instance, with the group of real substitutions
yz + 8'
where a, /?, 7, 8 are integers such that a8 — ^y = l, it is easy to see that, when
2-i a,nd Z2 are real, we can make the numerical magnitude of
jZi + 8 yz.2 4- 8
as small a non-evanescent quantity as we please by proper choice of oc, /3, 7, S :
thus the group is improperly discontinuous, because for real values of the
variable it admits infinitesimal transformations. But such infinitesimal
transformations are not possible, when z does not lie on the axis of real
quantities, that is, when z is complex : so that, for all complex values of
z, the group is properly discontinuous.
The various points, derived from a single point by linear substitutions,
will, in subsequent investigations, be found to be arguments of a uniform
function. Continuous groups would give a succession of points infinitely
close together; that is, for these points, either /(^) would be unaltered in
value for a line or a small area of points and therefore constant everywhere,
or else the point would be an essential singularity, as in § 37. We shall
therefore consider only discontinuous groups.
A group containing only a finite number of substitutions is easily seen to
be discontinuous : hence the groups which are to be considered in the present
connection are the discontinuous groups which arise from a finite number of
fundamental substitutions*.
The constants of all linear substitutions of the form -^ are sup-
posed subject to the relation ad — be = 1. This condition holds for all
combinations, if it hold for the components of the combination. For let
az + ^ rpi az + b
then ST =
7^ -I- S ' cz + d'
(aa + ^c)z^ab + ^d _ Az + B
{ya + 8c) z + yb + 8d Cz + L '
whence AD- BC = {a8 - ^y)(ad - bc)= I.
''* These discontinuous, or discrete, groups will be considered from the point of view of auto-
morphic functions. But the theory of such groups, which has many and wide applications quite
outside the range of the subject of this treatise, can be applied to other parts of our subject.
Thus it has been connected with the discussion of Riemann's surfaces by Dyck, Math. Ann.,
t. xvii, (1880), pp. 473—509, and by Hurwitz (I.e., p. 456, note).
282.] FINITE GROUPS • 7l9
It is easy to see that ST (= U) and TS (= V) are of the same class, that
is, they are elliptic, parabolic, hyperbolic or loxodromic together : but there is
no limitation on the class arising from the character of the component sub-
stitutions.
Moreover, if U = V, so that S and T are interchangeable, then
a — dch
oi^8 " 7 " ;s '
that is, *S* and T have the same fixed points. They can be applied in any
order ; and, for any given number of occurrences of 8 and a given number of
occurrences of T, the composite substitution will give the same point. Thus
if S = z+ CO, then T = 2 + Qi' ; it S = kz, then T = k'z. The class of func-
tions, which have their argument subjecj, to interchangeable substitutions
of the former category, have already been considered : they are the periodic
functions with additive periodicity. The group is S'^T'^', {= z + mw + mfo)'),
for all integral values of m and of in.
The latter class of functions have what may be called a factorial
periodicity, that is, they resume their value when the argument is mul-
tiplied by a constant *.
283. Some examples have already been given of groups containing a
finite number of substitutions!, in the case of certain periodic elliptic
substitutions. The effect of such substitutions is (p. 628) to change a
crescent-shaped part of the plane having its angles at the (conjugate) fixed
points of the substitution into consecutive crescent-shaped parts : and so to
cover, the whole plane in the passage of a substitution through the elements
constituting its period. They form the simplest discontinuous group — in
that they have only one fundamental substitution and only a finite number
of derived substitutions.
The groups which are next in point of simplicity are those with only
two substitutions that are fundamental and only a finite number that
are composite. Both of the fundamental substitutions must be periodic,
and therefore elliptic, by § 260. Taking one of these groups as an example,
* Functions having this property are discussed by Piucherle, " Sulle funzioni monodrome
aventi un' equazione caratteristica," Rend. 1st. Lomb., Ser. 2, t. xii, (1879), pp. 536 — 542. See
also Eausenberger's Tlieorie der periodischen Functionen, (Leipzig, Teubi)er, 1884) : in particular,
Section VI.
t The complete theory of finite groups of linear substitutions is discussed, partly in its
geometrical lelation with polyhedral functions, by Klein, Math. Ann., t ix, (1876), pp. 183—188,
and, in its algebraical aspect, by Gordan, Math. Ann., t. xii, (1877), pp. 23 — 46. A reference
to these memoirs will shew that the previous chapter contains all the essentially distinct finite
groups of linear substitutions.
720 EXAMPLE OF FINITE GROUP [288.
one of its fundamental substitutions has ± 1 as its fixed points and it is
periodic of the second order : it is evidently
o 1
w = oz = - .
z
The other has ^ and oo as its fixed points, and it is periodic of the second
order : it is evidently
xu=Tz = \-z.
Evidently ^H = z, T'z = z, (S^S-\ T=T-^), so that we have already all the
powers of the fundamental substitutions taken separately.
But it is necessary to combine them. We have Uz = STz = , a new
substitution : and then
U'z = ^^, mz = z,
z
so that U is periodic of the third order. Again
Vz = TSz = ^-^,
z
which is not a new substitution, for Vz — U^z : and it is easy to see that there
is only one other substitution, which may be taken to be either TUz ov 8Vz:
it gives
TUz=SVz = -^^,
z — 1
again periodic of the second order.
Hence the group consists of the six substitutions for 2r«given by
1 1 z- 1 z
z \ — z z z—1
taking account of the identical substitution.
These finite discontinuous groups are of importance in the theory of
polyhedral functions : to some of their properties we shall return later.
Next, and as the last special illustration for the present, we form a
discontinuous group with two fundamental substitutions but containing an
infinite number of composite* substitutions. As one of the two that are
fundamental, we take
w=Tz=--,
z
which is elliptic and periodic of the second order. As the other, we take
w = Sz= z + 1,
which is parabolic and not periodic. All the substitutions are real.
One such group has already occurred : its fundamental (parabolic) substitutions were
w — Sz = z + o}, iv=Tz=z + w'.
283.] DIVISION OF PLANE 721
Evidently T^z = z, so that T = T~^ : and S'^z = z + m, where m is any
integer. Then all the composite substitutions are either of the form
...SpTS^'TS'^z or of the form ...SpTS^'TS^Tz, both of these being included
in -J , where a, b, c, d are integers, such that ad — be = 1.
CZ ~r a
Ex. Prove the converse— that the substitution -7 , where a, b, c, d are integers
such that ad — bc=l, is compounded of the substitutions JS and T.
This group, again, is of the utmost importance : it arises in the theory of
the elliptic modular-functions. As with the polyhedral groups, the general
discussion of the properties will be deferred : but it is advantageous to
discuss one of its properties now, because it forms a convenient introduction
to, and illustration of, the corresponding part of the theory of groups of
general substitutions.
284. In the discussion of the functions with additive periodicity, it was
found convenient to divide the plane into an infinite number of regions such
that a region was changed into some other region when to every point of the
former Avas applied a transformation of the form z + m&) + mfco\ that is, a
substitution : and the regions were so chosen that no two homologous points,
that is, points connected by a substitution, were within one region, and each
region contained one point homologous with an assigned point in any region
of reference.
Similarly, in the case when the variable is subject to the substitutions of
an infinite group, it is convenient to divide the plane into an infinite number
of regions ; each region is to be associated with a substitution which, applied
to the^points of a region of reference, gives all the points of the region, and
each region is to contain one and only one point derived from a given point
by the substitutions of the group. It is a condition that the complete plane
is to be covered once and only once by the aggregate of the regions.
When the discontinuous group has only the two fundamental substitutions,
Sz = z + 1 and Tz = , the division of the plane is easy : the difficulty of
z
determining an initial region of reference is slight, relatively to that which
has to be overcome in more general groups*.
The ordinates of z and w (= Sz) are positive together or negative together ;
and similarly for the ordinates of z and tu (= Tz) : so that it will suffice to
divide the half-plane on the positive side of the axis of real quantities.
For the repetitions of the substitution S, it is evidently sufficient to divide
the plane into a series of strips, bounded by straight lines parallel to the axis
of y at unit distance apart.
* In addition to the references already given, a memoir by Hurwitz, Math. Ann., t. xviii,
(1881), pp. 531—544, may be consulted for this group. ,
P. F. 46
72:2 DIVISION OF PLANE BY [284.
For the application of the substitution T, we have to invert with regard
to a circle of radius 1 and centre the origin, and to take the reflexion of the
inversion in the axis of y.
In these circumstances, we can choose as an initial region of reference, the
space bounded by the conditions
1 1 , , .
It is sufficient to prove that any point in this region when subjected to a
substitution of the group, necessarily of the form — -^ , where a, b, c, d are
integers such that ad — bc = l, is transformed to some point without the
region, and that the aggregate of the regions covers the half-plane.
If c be 0, then a=l =d and the transformation is only some power of *S',
which transforms the point out of the region.
If c be + 1, then, since ad — hc = 1, we have
1
z + d
a and d being integers. For any point z within the region, |i^ + cZ|, which is
the distance of the point from some point 0, ±1, ± 2, ... on the axis of x, is
> 1 : hence
jw — a| < 1,
that is, the distance of w from some point 0, +1, + 2, ... on the axis is < 1,
and therefore the transformed point is without the region.
Similarly, if c be — 1.
-rn , , -, ^ . all
11 c be > 1, then w — = —
c c- d'
z+-
c
d
z+-
c
VS.
^ 2 •
V3
2
a
w
c
As z is within the region, z +- ^-^ '• and therefore
1 1
^ c^ ^ 4 '
so that
Hence the distance of w from some point 0, + 1, + 2, ... on the axis of x
is < ^ \/S, that is, the transformed point is without the region.
The exceptions are points on the boundary of the region. The boundary
x = — ^ is transformed by S to x = + ^: the boundary x"^ -f ^Z"- = 1 is trans-
formed by T into itself: but all other points are transformed into others
without the region.
284.]
ELLIPTIC MODULAR-FUNCTION GROUP
723
We now apply the substitutions S and T to this region and to the
resulting regions. Each substitution is uniform and is reversible : so that
to a given point in the initial region there is one, and only one, point in each
other region.
The accompanying diagram (Fig. 108) gives part of the division of the
plane into regions, the substitutions associated with each region being
placed in the region in the figure ; it is easy to see that the aggregate of
regions completely covers the half-plane. All the linear boundaries of S'^,
for different integral values of n, are changed by the substitution T into
circles having their centres on the axis of x and touching at A : thus the
boundary between S and S'" is transformed into the boundary between
TS and TS^. All the lines which bound the regions are circles having
their centres on the axis of x or are straight lines perpendicular to that
axis ; and the configuration of each strip is the same throughout the
diagram.
Fig. 108.
It will be noticed that in one region there are two symbols, viz., S~^TS~^
and TST : the region can be constructed either by S'^ applied to TS-^ or by
T applied to ST. It therefore follows that
TST = S-'TS-\
Hence S . TST. S=S. S'^TS'' .S=T,
or, since T^ = 1 , we have S TST ST = 1 = TSTSTS,
a relation among the fundamental substitutions. Thus the symbol of any
region is not unique : and, as a matter of fact, if we pass clockwise in a small
46—2
724 FUCHSIAN GROUPS [284.
circuit round from the initial region, we find the regions to be 1, T, TS, TST,
TST8, TSTST, TST8TS, the seventh being the same as the first and giving
the above relation.
By means of this relation it will be found possible to identify the non-
unique significations of the various regions. At each point there are six
regions thus circulating always, either in the form ®S, ©ST, SSTS, ... or in
the form @T, @TS, @T8T,.... And by successive transformations, the space
towards the axis of on is distributed into regions.
The decision of the region to which a boundary should be assigned will
be made later in the general investigation ; it will prove a convenient step
towards the grouping of edges of a region in conjugate pairs.
Note. It may be proved in the same way that, for any discontinuous
group of substitutions, the plane of the variable can be divided into regions
of a similar character. As will subsequently appear, there is considerable
freedom of choice of an initial region of reference, which may be called a
fundamental region.
285. We now pass to the consideration of the more general discontinuous
groups, based on the composition of a finite number of fundamental substitu-
tions. By means of these groups and in connection with them, the plane of
the variable can be divided into regions, one corresponding to each substitu-
tion of the group. The regions are said to be congruent to one another :
the infinite series of points, one in each of the congruent regions, which arise
from z when all the substitutions of the group are applied to z, are said to
be corresponding or homologous points : and the point in Mq of the series is
the irreducible point of the series. As remarked before, the correspondence
between two regions is uniform : interiors transform to interiors, boundaries
to boundaries.
/ Two regions are said * to be contiguous, when a part of their boundaries is
commbri to both. Each region, lying entirely in the finite part of the plane,
is closed: the boundary is made up of a succession of lines whicb may for
convenience be called edges, and the meeting-point of two edges may for con-
venience be called a corner.
Such a group, when all the substitutions are real, is called f Fuchsian,
by Poincare ; the preceding example will furnish a simple illustration, useful
for occasional reference. All the substitutions are of the form
ds^ + bs . .
CgZ + ds '
* Poincare uses the term limitrophes.
t Math. Ann., t. xix, p. 554, t. xx, pp. 52, 53: Acta Math., t. i, p. 62. The same term is
applied to a less limited class of groups; see p. 740, note.
285;] CATEGORIES 725
which form will be denoted by fs{z). We shall suppose that an infinite
group of real substitutions is given, and that it is known independently to
be a discontinuous group: we proceed to consider the characteristic properties
of the associated division of the plane, which is to be covered once and only
once by the aggregate of the regions. The fundamental region is denoted
by R^: the region, which results when the substitution fm{z) is applied
to the points of Rq, will be denoted by Rm.
So long as we deal with real substitutions, it is sufficient to divide the
half-plane above the axis of x into regions : and this axis may be looked upon
as a boundary of the plane. Since the group is infinite, the division into
regions must extend in all directions in the plane to its finite or infinite
boundaries : for we should otherwise have infinitesimal transformations. Thus
the edge of a region is either the edge of a contiguous region, and then it is
said to be of the first hind ; or it is a part of the boundary of the plane, that
is, in the present case it is a part of the axis of x : and then it is said to be of
the second kind. Since all real substitutions transform a point above the axis
of X into another point above the axis of x, it follows that all edges congruent
with an edge of the first kind (an edge lying ofi" the axis of x) themselves
lie ofi" the axis of x, that is, are of the first kind : and similarly all edges con-
gruent with an edge of the second kind are themselves of the second kind.
The corners, being the extremities of the edges, are of three categories.
If a corner be an extremity of two edges of the first kind and not on the
axis of X, then it is of the first category : and the infinite series of corners
homologous with it are of the first category. If it be common to two
edges of the first kind and lie on the axis of x, then it is of the second
category: and the infinite series of corners homologous with it are of the
second category. If it be common to two edges, one of the first and one of
the second kind, it is of the third category ; of course it lies on the axis
of X and the infinite series of corners homologous with it are of the third
category. We do not consider two edges of the second kind as meeting :
they would, in such a case, be regarded as a single edge.
Each edge of the first kind belongs to two regions. We do not assign
such an edge to either of the regions, but we use this community of
region to range edges as follows. Let the edge be Bp, common to Rq
and Rp ; then, making the substitution inverse to fp (z), say fp~'^ (z), Rp
becomes Rq, Rq becomes R-p, and Ep becomes fp~'^(Ep), which is necessarily
an edge of the first kind and is common to the new regions R_p and R^,
that is, it is an edge of Rq. Let it be Ep : then Ep and Ep' may be
the same or they may be different.
If Ep and Ep be different, then we have a pair of edges congruent to
one another : two such congruent edges of the same region are said to be
conjugate. Since the substitutions are of the linear type, the correspondence
726 FUNDAMENTAL SUBSTITUTIONS [285.
being uniform, not more than one edge of a region can be conjugate with
a given edge of that region.
If Ep and Ep be the same, then the substitution transforms Ep into
itself: hence some point on Ep must be transformed into itself As the edge
is of the first kind so that the point is above the axis of X, the substitution
is elliptic and has this point as the fixed point of the substitution in
the positive half-plane. The two parts of Ep can be regarded as two
edges: and the common point as the corner, evidently of the first category.
Because the directions of the edges measured aAvay from the point are
inclined at an angle tt, it follows that the multiplier of the elliptic sub-
stitution is 6"^ or —1. An illustration of this occurs in the special
example of § 284, where the circular boundary of the initial region of
reference is changed into itself by the fundamental substitution wz = — \,
that is,
w — i z — i
w + i z + i'
Hence the edges of the first kind are even in number and can he arranged
in conjugate pairs.
Further, a point on an edge of the first kind is transformed into a
point on the conjugate edge — uniquely, unless the point be a corner, when
it belongs to two edges. Hence points on edges of the first kind other than
corners correspond in pairs.
An edge of the second kind is transformed into one of the second kind,
but belonging to a different polygon : there is no correspondence between
points on edges of the second kind belonging to the same polygon.
Each corner, as the point common to two edges, belongs to at least three
regions. As a point of one edge, it will have as its homologue an extremity
of the conjugate edge : as a point of another edge, it will have as its homologue
an extremity of the edge conjugate to that other : and these homologues may
be the same or they may be different. Hence several corners of a given
region may he homologous : the set of homologous coymers of a given region is
called a cycle. Since points of a series homologous with a given point all
belong to one category, it is convenient to arrange the cycles in connection
with the categories of the component parts.
The number of edges of the first kind is even, say 2?i : and they can be
arranged in pairs of conjugates, say E^, En+i ; Ez, En+2 ; • . . • Then since En+p
is the conjugate of Ep, and fn+p (z) is the substitution which changes Rq into
Rn+p, fn+p{z) is a substitution changing Ep into E^-^p. After the preceding
explanation, /j5~^ {z) is also a substitution changing Ep into its conjugate En+p :
hence we have
fn+p (^) =/p ^ i^)'
285.] FUNDAMENTAL SUBSTITUTIONS 72?
Hence for a division of the plane, each region of which has 2n edges of the
first kind, the group contains n fundamental substitutions : the remaining n
substitutions, necessary to construct the remaining contiguous regions, are
obtained by taking the first inverses of the fundamental substitutions.
The edge Ep has been taken as the edge common to Ro and Rp, the region
derived from R^ by the substitution fp (z). Every region will have an edge
congruent to Ep : if Ri be one such region, then the region, on the other side
of that line and having that line for an edge (the edge is, for that other
region, the congruent of the conjugate of Ep), is obtainable from Rq by the
substitution y^- 1/^(5)}. We thus have an easy method of determining the
substitution to be associated with the region, by considering the edges which
are crossed in passing to the region : and, conversely, when the substitutions
are associated with the regions, the correspondence of the edges is known.
As in the special example, there are relations among the fundamental
substitutions. The simplest mode of determining them is to describe a small
circuit round each corner of Rq in succession : in the description of the circuit,
the symbol of each new region can be derived by a knowledge of the edge last
crossed and when the circuit is closed the last symbol is the symbol also of R^,
so that a relation is obtained. «
286. The only limitations as yet assigned to the initial region (and there-
fore to each of the regions) of the plane are (i) that it contains only one point
homologous with z, and (ii) that the even number of edges of the first kind
can be arranged in congruent conjugate pairs. But now,
without detracting from the generality of the division, we q E___^_^
can modify the initial region in such a way that all the /^---il^ — -^
edges of the first kind are arcs of circles with their centres
on the axis of x. For let C. ..AFB. ..DGG be a region with \ \
CGD and AFB for conjugate edges; join CD by an arc of \ ^ \
a circle CED with its centre on the axis of x : and apply to aV^^^^^o^J^ B
CED the substitution inverse to that which gives the region ^
. . Fi". 109
in which E lies : let AHB be the result, being also (§ 258)
an arc of a circle with its .centre on the axis of x. Then the part AFBHA,
say So, is transformed to CGD EG, say So, by the substitution which causes a
passage from Rq across CGD into another region: every point in *S„ has a
homologue in >So' : and there is, by the hypothesis that Ro is the initial region,
no homologue in Ro of a point in So except the point itself. If, then, we take
away ^0 from Ro and add >S^o') we have a new region
Ro' = Ro + So — So-
It satisfies all the conditions which apply to the regions so far obtained : there
is no point in Ro homologous with a point in it, and the conjugate edges
CGD and AFB are replaced by conjugate edges CED, AHB congruent
728 CONVEXITY OF [286.
by the same substitution as the former pair. And the new conjugate
edges are circles having their centres on the axis of x.
Proceeding in this way with each pair of conjugate edges that are not
arcs of circles having their centres on the axis of x, and replacing it by a pair
of conjugate edges congruent by the same substitution and consisting of
arcs of circles having their centres on the axis of x, we ultimately obtain a
region in which all the edges of the first kind are arcs of circles having .their
centres on the axis of x. These can, of course, be arranged in conjugate pairs,
congruent by the assigned fundamental substitutions. Straight lines perpen-
dicular to the axis of x count as circles with centres at x= 2.
289.] EXAMPLE 735
The fundamental substitution, which changes AD into CZ), has D and the complex
conjugate to B for its fixed points: these points are +^psin — . The argument of the
9
multiplier is — , being the angle ADC: hence the substitution is
w — ip Sin — z — ip sin — ^
= e
which reduces to
w + ip sin— z + ip sin —
VI '^ Vfl
z cos — Vp sin^ —
w —
1-cos —
p v%
where p has the value given by the above equation.
This substitution, and the substitution w= — , are the fundamental substitutions of
z
the group. The special illustration in § 284 gives
m = 00 , p = 00 , ?i = 3, p sin^ — = 2 cos — = 1 :
VI 11
the special form therefore is
w=z + \.
Taking cos— = a, cos- = 6, /S. = (a'^ + b'^ — \)^, we have p (l-a^) = h + ^\ the second
VI 11 r \ /
fundamental substitution is
az+A + ft
i = Sz=
(A-6)3 + a"
It is easy to see that
where Tz= — ; the complete figure can be constructed as in § 284.
An interesting figure occurs for m=4, n = Q.
In the same way it may be proved that, if an elliptic substitution have re * for its
common points and 29 for the argument of its multiplier, its expression is
__ Az+B
^~Gz + D'
, , sin(<9-e) „ sine ^ 1 sin 6 ^ sin((9 + e)
where A = ^ — -^-^ , B=r—. — ^r , C= -. — ^, D= ^ — ^ ,
sm 6 sm ^ /• sm ^ sm 6
Taking now the more general case where B = ^ri D = — , A + C= — , let B (in
figure 112) be the point 6e^\ and A the point ae*'. Then the substitution which transforms
AB into BC is the above, when ^ = /3, r = h, e=B, so that, if C be ce^*.
yj g sin (/3-^)e°'' + & sing
giving two relations among the constants.
■^sin5e"' + sin(/3 + 5)
736 FUNDAMENTAL [289.
Similarly, two more relations will arise out of the substitution which transforms CD
into DA. And three relations are given by the conditions that the sum of the angles at
A and C is an aliquot part of Stt, and that each of the angles B and D is an aUquot part
of 27r.
290. All the substitutions hitherto considered have been real : we now
pass to the consideration of those which have complex coefficients. Let
75 + S
be such an one, supposed discontinuous : then the effect on a point is obtained
by displacing the origin, inverting with respect to the new position, reflecting
through a line inclined to the axis of a; at some angle, and again displacing
the origin. The displacements of the origins do not alter the character of
relations of points, lines, and curves : so that the ' essential parts of the
transformation are an inversion and a reflexion.
Let a group of real substitutions of the character considered in the
preceding sections be transformed by the foregoing single complex substitu-
tion : a new group
az + ^ 70 + S
+ 6
we have p^ = zzq, and therefore
/ ^ ^■^00:70 + Zq^jo + zaSp + /38o _ az + ^
^^o77o + ^o7oS + zy^o + 8S0 yz + B'
on the removal of the factor 70^0 + ^o common to the numerator and the
denominator; and ^' vanishes when ^=0. The uniqueness of the result is
an a posteriori justification of the initial assumption that one and the same
point Q is derived from P, whatever be the inversions that are equivalent to
the linear substitution.
Ex. 1. Let an elliptic substitution have ti and v as its fixed points.
Draw two circles in the plane, passing through u and v and intersecting at an angle
equal to half the argument of the multiplier. The transformation of the plane, caused by
the substitution, is equivalent to inversions at these cii'cles ; the corresponding transforma-
tion of the space above the plane is equivalent to inversions at the spheres, having these
circles as equatorial circles. It therefore fallows that every point on the line of intersection
of the spheres remains unchanged : hence when a Kleinian substitution is elliptic, evet'y
point on the circle^ in a plane perpendicular to the plane of x, y and having the line joining
the common points of the substitution as its diameter, is unchanged by the substitution.
Poincare calls this circle C the double (or fixed) circle of the elliptic substitution.
. Ex. 2. Prove that, when a Kleinian substitution is hyperbolic, the only points in
space, which are unchanged by it, are its double points in the plane of x, y ; and shew
that it changes any circle through those points into itself and also any sphere through
those points into itself.
294.] KLEINIAN GEOUPS 747
Ex. 3. • Prove that, when the substitution is loxodromic, the circle C, in a plane
perpendicular to the plane x, y and having as its diameter the line joining the common
points of the substitution, is transformed into itself, but that the only points on the
circumference left unchanged are the common points.
Ex. 4. Obtain the corresponding properties of the substitution when it is parabolic.
(All these results are due to Poincare.)
295. The process of obtaining the division of the ^^-plane by means of
Kleinian groups is similar to that adopted for Fuchsian groups, except
that now there is no axis of real quantities or no fundamental circle
conserved in that plane during the substitutions : and thus the whole
plane is distributed. The polygons will be bounded by arcs of circles as
before : but a polygon will not necessarily be simply connected. Multiple
connectivity has already arisen in connection with real groups of the third
family by taking the plane on both sides of the axis.
As there are no edges of the second kind for polygons determined by
Kleinian groups, the only cycles of corners of polygons are closed cycles ;
let A^, A-^, ..., An-i in order be such a cycle in a polygon R^. Round Aq
describe a small curve, and let the successive polygons along this curve be
Ra, R^, ..., Rn-1, Rn, •••• The corner ^o belongs to each of these polygons:
when considered as belonging to R^, it will in that polygon be the homologue
of Am as belonging to R^, if m< n; but, as belonging to Rn, it will, in that
polygon, be the homologue of Aq as belonging to R^. Hence the substitution,
which changes Ra into Rn, has Aq for a fixed point.
This substitution may be either elliptic or parabolic, (but not hyperbolic,
I 292): that it cannot be loxodromic may be seen as follows. Let pe*" be
the multiplier, where (§ 259) p is not unity and a is not zero : and let
2o denote the aggregate of polygons Rq, R^, ..., Rn-i, 2i the aggregate
Rn, ..., Rm-i, and so on. Then Xo is changed to 2i, 2i to Sa, and so on,
by the substitution. Let p be an integer such that ptw ^ 27r ; then, when
the substitution has been applied p times, the aggregate of the polygons
is 1p, and it will cover the whole or part of one of the aggregates 2o, Sj, ....
But, because p^ is not unity, Xp does not coincide with that aggregate or the
part of that aggregate : the substitution is not then properly discontinuous,
contrary to the definition of the group. , Hence there is no loxodromic
substitution in the group. If the substitution be elliptic, the sum of the
angles of the cycle must be a submultiple of 27r ; when it is parabolic, each
angle of the cycle is zero.
In the generalised equations whereby points of space are transformed
into one another, the plane of x, y is conserved throughout : it is natural
therefore to consider the division of space on the positive side of this plane
into regions Po, Pi, ..., such that Pq is changed into all the other regions in
turn by the application to it of the generalised equations. The following
748 DIVISION OF SPACE [295.
results can be obtained by considerations similar to those before adduced in
the division of a plane*.
The boundaries of regions are either portions of spheres, having their
centres in the plane of x, y, or they are portions of that plane : the
regions are called polyhedral, and such boundaries are called faces. If the
face is spherical, it is said to be of the first kind : if it is a portion of
the plane of oc, y, it is said to be of the second kind. Faces of the
second kind, being in the plane of x, y and transformed into one another,
are polygons bounded by arcs of circles.
The intersections of faces are edges. Again, an edge is of the first
kind, when it is the intersection of two faces of the first kind : it is of
the second kind, when it is the intersection of a face of the first kind
with one of the second kind. An edge of the second kind is a circular
arc in the plane oi x, y : an edge of the first kind, being the intersection
of two spheres with their centres in the plane of x, y, is a circular arc,
which lies in a plane perpendicular to the plane of x, y and has its
centre in that plane.
The extremities of the edges are corners of the polyhedra. They are
of three categories :
(i) those which are above the plane of x, y and are the common
extremities of at least three edges of the first kind :
(ii) those which lie in the plane of x, y and are the common extremities
of at least three edges of the first kind :
(iii) those which lie in the plane of x, y and are the common extremities
of at least one edge of the first kind and of at least two edges of
the second kind.
Moreover, points at which two faces touch can be regarded as isolated corners,
the edges of which they are the intersections not being in evidence.
Faces of a polyhedron, which are of the first kind, are conjugate in pairs:
two conjugate faces are congruent by a fundamental substitution of the group.
Edges of the first kind, being, the limits of the faces, arrange themselves
in cycles, in the same way as the angles of a polygon in the division of the
plane. If E^, E^, ..., En-i be the n edges in a cycle, the number of regions
which have an edge in Eo is a multiple of n : and the sum of the dihedral
angles at the edges in a cycle (the dihedral angle at an edge being the
constant angle between the faces, which intersect along the edge) is a
submultiple of 27r.
The relation between the polyhedral divisions of space and the polygonal
divisions of the plane is as follows. Let the group be such as to cause the
* See, in particular, Poincar^, Acta Math., t. iii, pp. 66 et seq.
295.] FUNDAMENTAL POLYHEDRA AND POLYGONS 749
fundamental polyhedron Pq to possess n faces of the second kind, say ^oi,
^02, •••, F^n- Every congruent polyhedron will then have n faces of the
second kind; let those of Pg be jP^i, -^82, •••, Fm- Every point in the plane
of X, y belongs to some one of the complete set of faces of the second kind :
and, except for certain singular points and certain singular lines, no point
belongs to more than one face, for the proper discontinuity of the group
requires that no point of space belongs to more than one polyhedron.
Then the plane of x, y is divided into n regions, say Z)i, Dg, •••, Dn', each
of these regions is composed of an infinite number of polygons, consisting of
the polygonal faces F. Thus D,. is composed of F^^., F-^^, F.^r, ... ; and these
polygonal areas are such that the substitution Sg transforms -For into Fg^.
Hence it appears that, by a Kleinian group, the whole plane is divided into
a finite number of regions ; and that each region is divided into an infinite
number of polygons, which are congruent to one another by the substitutions
of the group.
296. The preceding groups of substitutions, that have complex co-
eflficients, have been assumed to be properly discontinuous.
JSx. Prove that, if any group of substitutions with complex coefficients be improperly
discontinuous, it is improperly discontinuous only for points in the plane of x, y.
(Poincare.)
One of the simplest and most important of the improperly discontinuous
groups of substitutions, is that compounded from the three fundamental
substitutions
z =Sz = z-^\, z =-Tz = , z' =Yz = z + %,
z
where % has the ordinary meaning. All the substitutions are easily proved to
be of the form
a^ + /3
7^ -f- S '
where aS — ^^7 = 1, and a, y8, 7, S are complex integers, that is, are represented
by m + ni, where m and n are integers. This is the evident generalisation of
the modular-function group: consequently there is at once a suggested
generalisation to a polyhedron of reference, bounded by
which will thus have one spherical and four (accidentally) plane faces.
The following method of consideration of the points included by the
polyhedron of reference differs from that which was adopted for the polygon
of reference in the plane.
If possible, let a point (|, 77, ^) lying within the above region be transformed
by the equations generalised from some one substitution of the group, say
750 EXAMPLE OF AN IMPROPERLY [296.
from " ^ \, , into another point of the region, say ^', rj', ^'. Then we have
l>-i> 4>^>-i, r + ^^ + r>i-
From the last, it follows that ^ > ^^ : and similarly for |', r]', ^', by the
\/ z
hypothesis that the point is in the region. Now
^' 1 1
and therefore 1/(?D = l7p + p |7^ + ^P-
Hence, as ^ and ^' are both > -7^ , we have |7|^ < 2 : so that, because 7 is
a complex integer, we have
7 = 0, + 1, + i
as the only possible cases.
If ry = 0, then since aS — ^y — 1, we have a8 = 1 and a, S are complex
integers : thus either
a=l) a=:_l] a= i] a — — i
For the first of these sub-cases we have, from the equations of the substitu-
tion,
where yS is a complex integer : if the new point lie within the region, then
/3 = 0, and we have
z' = z, K'=t
which is merely an identity.
For the second, we have z = z — ^ : leading to the same result.
For the third, we have, since So = i,
z = — z-\- 1/3.
But as j I' 1 , 1 7;' 1 , 11^ 1 , 1 77 1 are all less than \, we have /3 = 0, and so
f = -|, v' = -v; and ^'=C
For the fourth case, we have
z' = — z — i^,
leading to the same result as the third. Hence, if 7 = 0, the only point lying
within the region is given by
determined by the substitution w' = — . , which is TVT~^ V~^TV.
296.] DISCONTINUOUS GROUP 751
If I7I = 1, that is, 77o= 1, then
, p = p'' + 2;oyo8 + zyBo + 8So.
Of the two quantities ^ and f ', one will be not greater than the other : we
choose ^ to he that one and consider the accordingly associated substitution*.
Thus ^/r^l, p2>l, andso
^o7oS 4- ^7^0 + 5^0 < 0,
say ^„l + ^io^Sao^()_
7 To 7 7o
Now I7I = 1, so that - is of the form p + iq, where p and q are integers : thus
Ave have
p'' + q^+ 2p^ + 2qr) < 0,
which is impossible because 2f < 1, 2?; < 1.
Hence it follows that within the region there are only two equivalent
points, derived by the generalised equations from the substitution
f %w
w = -.
— I
and that all points within the region can be arranged in equivalent pairs
^, 77, ^ and -^,-v,^-
If the region be symmetrically divided into two, so that the boundaries of
a new region are
then no point within the new region is equivalent to any other point in the
regionf. As in the division of the plane by the modular group, it is easy
to see that the whole space above the plane of ^, ij is divided by the group :
therefore the region is a polyhedron of reference for the group composed of the
fundamental substitutions S, T, V.
The preceding substitutions, with complex integers for coefficients, are of use in appli-
cations to the discussion of binary quadratic forms in the theory of numbers. The special
division of all space corresponds, of course, to the character of the coefficients in the
substitutions : other divisions for similar groups are possible, as is proved in Poincare's
memoir already quoted.
* Were it f ', all that would be necessary would be to take the inverse substitution,
t Bianchi, Math. Ann., t. xxxviii, (1891), pp. 313—324, t. xl, (1892), pp. 332—412; Picard, ib ,
t. xxxix, (1891), pp. 142—144; Mathews, Quart. Journ. Math., vol. xxv, (1891), pp. 289—296.
752 EXAMPLE [296.
These divisions all presuppose that the group is infinite : but similar divisions for only-
finite groups (and therefore with only a finite number of regions) are possible. These are
considered in detail in an interesting memoir by Goursat* ; the transformations conserve
an imaginary sphere instead of a real plane as in Poincare's theory.
Ex. Shew that, for the infinite group composed of the fundamental substitutions
z = , 2=0 + 1, z=z + e,
z
where e is a primitive cube root of unity, a fundamental region for the division of space
above the plane of z, corresponding to the generalised equations of the group, is a sym-
metrical third of the polyhedron extending to infinity above the sphere
and bounded by the sphere and the six planes
2£=±1, | + W3=±1, |-W3=±1- (Bianchi.)
* " Sur les substitutions orthogonales et les divisions regulieres de I'espace," Ann. de VEc.
Norm. Sup., 3'"'= Ser., t. vi, (1889), pp. 9—102. See also Schonflies, Math. Ann., t. xxxiv, (1889), ,
pp. 172 — 203 : other references are given in these papers.
CHAPTEE XXII.
AuTOMORPHic Functions. ,
297. As was stated in the course of the preceding chapter, we are
seeking the most general form of the arguments of functions which secures
the property of periodicity. The transformation of the arguments of trigo-
nometrical and of elliptic functions, which secures this property, is merely a
special case of a linear substitution : and thus the automorphic functions to
be discussed are such as identically satisfy the equation
where Si is any one of an assigned group of linear substitutions of which only
a finite number are fundamental.
Various references to authorities will be given in the present chapter, in connection
with illustrative examples of automorphic functions : but it is, of course, beyond the scope
of the present treatise, deahng only with the generalities of the theory of functions, to
enter into any detailed development of the properties of special classes of automorphic
functions such as, for instance, those commonly called polyhedral and those commonly
called elliptic-modular. Automorphic functions, of types less special than those just
mentioned, are called Ftichsian functions by Poincare, when they are determined in
association with a Fuchsian group of substitutions, and Kleinian functions, when they
are determined in association with a Kleinian group : as our purpose is to provide only
an introduction to the theory, the more general term automorphic will be adopted.
The establishment of the general classes of automorphic fimctions is eflfected by
Poincare in his memoirs in the early volumes of the A eta Mathematica, and by Klein in
hie memoir in the 21st volume of the Mathematische Annalen : these have been already
quoted (p. 716, note) : and Poincare gives various historical notes* on the earlier scattered
occurrences of automorphic functions and discontinuous groups. Other memoirs that may
be consulted with advantage are those of Von Mangoldtt, Weber |, Schottky§, Stahl||,
* Acta Math., t. i, pp. 61, 62, 293: ib., t. iii, p. 92. Poincare's memoirs occur in the first,
third, fourth and fifth volumes of this journal : a great part of the later memoirs is devoted to
their application to linear differential equations.
t Gbtt. Nadir., (1885), pp. 313—319; ib., (1886), pp. 1—29.
J Gbtt. Nachr., (1886), pp. 359—370.
§ Crelle, t. ci, (1887), pp. 227—272.
II Math. Ann., t. xxxiii, (1889), pp. 291—309.
F. F. 4S
754 ANHARMONIC GROUP 'AND FUNCTION [297.
Schlesinger* and Rittert : and there are two by BurnsideJ, of special interest and
importance in connection with the third of the seven families of groups (§ 292). Finally,
reference may be made to the comprehensive treatise** by Fricke and Klein.
298. We shall first consider functions associated with finite discrete
groups of linear substitutions.
There is a group of six substitutions
1 1 2-1 ^_
z \ — z z z—\
which (§ 283) is complete. Forming expressions z — x, z , z — ^Y—x),
1 QC *^ 1 OC
z — , z , z z. and multiplying them together, we can express
X ~" *ju OG 3G "^ ±.
their product in the form
(,-> _ ,y [ (^^-^+1)^ _ {x^-x + m
^ ^ \ {z'-zf {a?-xy I'
so that A {z) = ^-— -^
{z^ — zf
is a function of z which is unaltered by any of the transformations of its
variable given by the six substitutions of the group. The function is well
known, being connected with the six anhannonic ratios of four points in a
line which can all be expressed in terms of any one of them by means of the
substitutions.
Another illustration of a finite discrete group has already been furnished
in the periodic elliptic transformation of § 258, whereby a crescent of
the plane with its angle a subijiultiple of 27r was successively transformed,
ultimately returning to itself: so that the whole plane is divided into portions
equal in number to the periodic order of the substitution.
If a stereographic projection of the plane be made with regard to any
external point, we shall have the whole sphere divided into a number of
triangles, each bounded by two small circles and cutting at the same angle.
By choice of centre of projection, the common corners of the crescents can be
projected into the extremities of a diameter of the sphere : and then each of
the crescents is projected into a lune. The effect of a substitution on the
crescent is changed into a rotation round the diameter joining the vertices
of a lune through an angle equal to the angle of the lune.
299. This is merely one particular illustration of a general correspondence
between spherical rotations and plane homographies, as we now proceed to
shew. The general correspondence is based upon the following proposition
due to Cayley : —
* CreMe, t. cv, (1889), pp. 181—232.
t Math. Ann., t. xli, (1892), pp. 1—82.
X Loud. Math. Soc. Proc, vol. xxiii, (1892), pp. 48—88, ib., pp. 281—295.
** Vorlesungen ilber die Theorie der automorphen Functionen, (Leipzig, Teubner, Bd. i, 1897).
299.] HOMOGRAPHY AND ROTATIONS 755
When a sphere is displaced hy a rotation round a diameter, the variables of
the stereographic projections of any point in its original position and in its
displaced position are connected hy the relation
, _{d-\- ic) z — {h — ia)
{b 4- ia) z + (d - ic) '
where a, b, c, d are 7'eal quantities.
Rotation about a given diameter through an assigned angle gives a
unique position for the displaced point: and stereographic projection, which
is a conformal operation in that it preserves angles, also gives a unique point
as the projection of a given point. Hence taking the stereographic projec-
tion on a plane of the original position and the displaced position of a point
on the sphere, they will be uniquely related : that is, their complex variables
are connected by a lineo-linear relation, which thus leads to a linear substitu-
tion for the plane-transformation corresponding to the spherical rotation.
Now the extremities of the axis are unaltered by the rotation ; hence the
projections of these points are the fixed points of the substitution. If the
points be ^, ??, ^ and — f , — 77, — ^, on a sphere of radius unity, and if the
origin of projection be the north pole of the sphere, the fixed points of the
substitution are
^^^''^ and -l+h-
so that the substitution is of the form
, ^ + ir] ^ + ir)'
To determine the multiplier K, we take a point P very near C, one extremity
of the axis : let P' be the position after the rotation, so that CF' = CP. Then,
in the stereographic projection, the small arcs which correspond to OP and
CP' are equal in length, and they are inclined at an angle a. Hence the
multiplier K is e^'* : for when z, and therefore z', is nearly equal to — ■ -, y , a
fixed point of the substitution, the magnification is | ^ | and the angular
displacement is the argument of K, which is a.
Inserting the value of K, solving for z and using the condition
^^ + 7}^ + ^^= 1, we have
, _ {d + ic) z — (b- ia)
~ {b + ia) z + (d- ic) '
where a = |^ sin |a, b = rj sin \a, c= ^ sin -|a, d = cos ^a,
so that a^ + ¥ + c'' + d^ = 1,
the equivalent of the usual condition to which the four coefficients in any
48—2
756 HOMOGENEOUS SUBSTITUTIONS [299.
linear substitution are subject : it is evident that the substitution is elliptic.
The proposition* is thus proved.
When the axis of rotation is the diameter perpendicular to the plane, we
have, by § 256,
z = ke-^+i^, z' = A;e-^+^<*+»',
so that / = ^e*",
agreeing with the above result by taking ^=0 = 77, ^=1, so that a = = 6,
c = sin |a, d = cos ^a.
It should be noted that the formula gives two different sets of coefficients
for a single rotation : for the effect of the rotation is unaltered when it is
increased by 27r, a change in a which leads to the other signs for all the
constants a, b, c, d.
It thus appears that the rotation of a sphere about a diameter interchanges
pairs of points on the surface, the stereographic projections of which on the
plane of the equator are connected by an elliptic linear substitution : hence,
in the one case as in the other, the substitution is periodic when a, the
argument of the multiplier and the angle of rotation, is a submultiple of 27r.
In the discussion of functions related in their arguments to these linear
substitutions, it proves to be convenient to deal with homogeneous variables,
so that the algebraic forms which arise can be connected with the theory of
invariants. We take zz^ = z^ : the formulae of transformation may then be
represented by the equations
z^ = K {az^ + ^z^, z^ = K {'yz^ + hz^,
for the substitution z' = {az + ,8)/(yz + 8). As we are about to deal with
invariantive functions of position dependent upon rotations, it is important
to have the determinant of homogeneous transformation equal to unity.
This can be secured only if k = + 1 or if « = — 1 : the two values correspond
to the two sets of coefficients obtained in connection with the rotation.
Hence, in the present case, the formulae of homogeneous transformation are
z-i' =^{d + ic) Zi — {b — ia) z^, z^ = (6 + ia) z-^-\-id — ic) z^,
where a^ + 6^ + c^ + rf^, being the determinant of the substitution, = 1 ; every
rotation leads to two pairs of these homogeneous equations f. Each pair of
equations will be regarded as giving a homogeneous substitution.
Moreover, rotations can be compounded : and this composition is, in the
analytical expression of stereographically projected points, subject to the same
algebraic laws as is the composition of linear substitutions. If, then, there
* Cayley, Math. Ami., t. xv, (1879), pp. 238—240; Klein's Vorlesungen ilber das Ikosaeder,
pp. 32—34.
+ The succeeding account of the polyhedral functions is based on Klein's investigations, which
are collected in the first section of his Vorlesungen ilber das Ikosaeder (Leipzig, Teubner, 1884): see
also Cayley, Camb. Phil. Trans., vol. xiii (1883), pp. 4—68; Coll. Math. Papers, vol. xi, pp. 148—216.
It will be seen that the results are intimately related to the results obtained in §§ 271 — 279,
relative to the eonformal representation of figures, bounded by circular arcs, on a half-plane.
299.] GROUPS FOR THE REGULAR SOLIDS 757
be a complete group of rotations, that is, a group such that the composition
of any two rotations (including repetitions) leads to a rotation included in the
group, then there will be associated with it a complete group of linear
homogeneous substitutions. The groups are finite together, the number of
members in the group of homogeneous substitutions being double of the
number in the group of rotations : and the substitutions can be arranged in
pairs so that each pair is associated with one rotation.
300. Such groups of rotations arise in connection with the regular solids.
Let the sphere, which circumscribes such a solid, be of radius unity : and let
the edges of the solid be projected from the centre of the sphere into arcs of
great circles on the surface. Then the faces of the polyhedron will be repre-
sented on the surface of the sphere by closed curvilinear figures, the angular
points of which are summits of the polyhedron. There are rotations, of proper
magnitude, about diameters properly chosen, which displace the polyhedron
into coincidence (but not identity) with itself, and so reproduce the above-
mentioned division of the surface of the sphere : when all such rotations have
been determined, they form a group which may be called the group of the
solid. Each such rotation gives rise to two homogeneous substitutions, so
that there will thence be derived a finite group of discrete substitutions :
and as these are connected with the stereographic projection of the sphere,
they are evidently the group of substitutions which transform into one
another the divisions of the plane obtained by taking the stereographic
projection of the corresponding division of the surface of the sphere. For
the construction of such groups of substitutions, it will therefore be sufficient
to obtain the groups of rotations, considered in reference to the surface of
the sphere.
I. The Dihedral Group. The simplest case is that in which the solid,
hardly a proper solid, is composed of a couple of coincident regular polygons
of n sides* : a reference has already been made to this case. We suppose the
polygons to lie in the equator, so that their corners divide the equator into
n equal parts : one polygon becomes the upper half of the spherical surface,
the other the lower half The two poles of the equator, and the middle
points of the n arcs of the equator, are the corners of the corresponding solid.
Then the axes, rotations about which can bring the surface into such
coincidence with itself that its partition of the spherical surface is topo-
graphically the same in the new position as in the old, are
(i) the polar axis,
(ii) a diameter through each summit on the equator,
(iii) a diameter through each middle point of an edge :
the last two are the same or are different according as n is odd or is even.
* The solid may also be regarded as a doable pyramid.
758 DIHEDEAL GROUP AND FUNCTION [300.
For the polar axis, the necessary angle of rotation is an integral multiple
27-
of — . Thus we have ^ = = v, ^=1, and therefore ■
n
a = = 6, c = sm - , ct = cos — ,
n n
the substitutions are
iirr JTT
Z-i = 6 Zi, Z^ = 6 Z<^ ,
for r = 0, 1, ...,n — 1, and
iirr iirr
z(r= — e'^z^, %' = — e "•^2,
for the same values of r. These are included in the set
inr inr
2r/=e^^i, zl = e "^ z^,
forr = 0, 1, 2, ..., 2^1 — 1, being ^n in number: the identical substitution is
included for the same reason as before, when we associated a region of
reference in the ^r-plane with the identical substitution.
For each of the axes lying in the equator, the angle of rotation is
evidently tt. Let an angular point of the polygon lie on the axis of |, say at
I = 1, 77 = 0, ^=0. Then so far as concerns (ii) in the above set, if we take
the axis through the (r + l)th angular point, we have | = cos , 77 = sin ,
^= ; hence, as a is equal to tt, we have, for the corresponding substitutions,
z^' = ie "^ ^2, z^= ie '^ z^,
for r = 0, 1, . . ., n — 1, and
2nri 2rTci
z-i = — ie "■ Z2, Z2 = — ie '^ z^,
for the same values of r-
And so far as concerns (iii) in the above set, if we take an axis through
the middle point of the rth side, that is, the side which joins the rth and the
(2r-l)7r . (2r-l)7r
(r + l)th points, then ^ = cos ^ , 77 = sm — , ^= : hence as a
is equal to tt, we have, for the corresponding substitutions,
{2r-l)ni (2r-l) Tri
Zj'=ie *" Z2, Z2=ie ** z^,
for r = 0, 1, . . ., 71 — 1, and
(2r-l)Tri {.2r-l)m
Zi = — ie ^ Zz, Z2 = — ie '^ Z:^,
for the same values of r.
If n be even, the set of substitutions associated with (ii) are the same in
pairs, and likewise the set associated with (iii) ; if w be odd, the set associated
with (ii) is the same as the set associated with (iii). Thus in either case there
are 2n substitutions : and they are all included in the form
VKT
%Tvr
Z^ ^ '2'^ -^2?
■^2 ^~ "^^ ^\ 3
rr-O. 1, .
.., 2n-l.
300:]
TETRAHEDRAL GROUP
759
Thus the whole group of 4w substitutions, in their homogeneous form, is
■ e '" ^1
Zn ^^ 6 Zf}
zirr
z( = ie ^ z^
_77rr
for r = 0, . . . , 2w — 1 : and in the non-homogeneous form, the group is
z' = e~>^ z.
e~n'
Z =
where r = 0, 1, ..., n—\ for each of them. The non-homogeneous expres-
sions are not in their normal form in which the determinant of the coefficients
in the numerator and denominator is unity. Each expression gives two
homogeneous substitutions.
It is easy geometrically to see that all the axes have been retained : and
that they form a group, that is, composition of rotations about any two of the
axes is a rotation about one of the axes. The period for each of the equatorial
axes is 2 ; the period for a rotation about the polar axis depends on the
r
reducibilitv of - .
"^ n
Before passing to the construction of the functions which are unaltered
for the dihedral group of substitutions, we shall obtain the tetrahedral group
and construct the tetrahedral functions, for the explanations in regard to the
dihedral functions arise more naturally in the less simple case.
II. The Tetrahedral Group. We take a regular cube as in the figure.
Then ABGD is a tetrahedron, A'B'C'D' is the polar tetrahedron.
Fig. 119.
It is easy to see that the axes of rotation for the tetrahedron are
(i) the four diagonals of the cube AA', BB', CC, DD' ;
%^ TETRAHEDRAL GROUP - [300.
(ii) the three lines joining the middle points of the opposite edges of
the tetrahedron.
The latter pass through the centre of the cube and are perpendicular to
pairs of opposite faces. When the sphere circumscribing the cube is drawn,
the three axes in (ii) intersect the sphere in six points which are the angles
of a regular octahedron. Thus, though the axes of rotation for the three
solids are not the same, the tetrahedron, the cube, and the octahedron may
be considered together: in fact, in the present arrangement whereby the
surface of the sphere is considered, the cube is merely the combination of the
tetrahedron and its polar.
For each of the diagonals of the cube, the necessary angle of rotation
for the tetrahedron is or f tt or |7r : the first of these gives identity, and
the others give two rotations for each of the four diagonals of the cube, so
that there are eight in all.
For each of the diagonals of the octahedron, the angle of rotation for
the tetrahedron is tt : there are thus three rotations.
With these we associate identity. Hence the number of rotations for the
tetrahedron is (8 + 3 + 1 =) 12 in all.
There are two sets of expressions for the tetrahedron according to the
position of the coordinate axes of the sphere. One set arises when these are
taken along Ox, Oy, Oz, the diagonals of the octahedron; the other arises
when a coordinate plane is made to coincide with a plane of symmetry of the
tetrahedron such as B'DBD'.
Let the axes be the diagonals of the octahedron. The results are
obtainable just as before, and so may now merely be stated:
For OB', ^ = V=^=^^'> when a = f7r, the substitution is
, _ z + i
~ z — i'
and when a = ^ir, the substitution is
, .z + \
z =1 ^ .
z — \
For OA, ^= — 7] = ^= --; when a = |7r, the substitution is
.z + 1
and when a = |7r, the substitution is
, z-i
z = -. .
Z + l
300.] OF SUBSTITUTIONS 761
For OG, — ^ = '7 = ?=-7q5 when a = |7r, the substitution is
z =% ,
z + l
and when a. = ^tt, the substitution is
, z + i
z = -. .
z — t
For OD', — ^ = — '? = ^= -7^ ; when a = |7r, the substitution is
, z — i
z + i'
and when a = ^ir, the substitution is
.z-1
Z =-l -.
z+1
For Ox, 1 = 1, 'n = (), ^=0 and a = tt : the substitution is
Z
For Oy, f = 0, ■/7 = 1, ^=0, and a = tt : the substitution is
1
z = .
z
For Oz, f = 0, 7? = 0, ^=1 and a = tt : the substitution is
z' = -z.
And identity is ^' = z.
Hence the group of tetrahedral non-homogeneous substitutions is
, 1 .z—1 .z + l z — i z + i
z = -^ z, +-, +z =■ , + I =- , + — --. , ± . ,
- z - z+l z-1 z + l z-%
when the axes of reference in the sphere are the diameters bisecting opposite
edges of the tetrahedron. Each of these substitutions gives rise to two homo-
geneous substitutions, making 24 in all.
To obtain the transformations in the case when the plane of xz is a plane
of symmetry of the tetrahedron passing through one edge and bisecting the
opposite edge, such as B'DBD' in the figure, it is sufficient to rotate the
preceding configuration through an angle ^tt about the preceding O^f-axis,
and then to construct the corresponding changes in the preceding formulae.
For this rotation we have, with the preceding notation of § 299, ^ = = ?;,
^=1, a = i7r: then a = = b, c = sini7r, c? = cosi7r, so that d + ic = e^^'^^ :
and therefore the ^' of the displaced point in the stereographic projection is
connected with the ^ of the undisplaced point in the stereographic projection
by the equation
y,_d + ic in_l+i^y
762 TETRAHEDRAL SUBSTITUTIONS [300.
If then Z be the variable of the projection of the undisplaced point and Z'
that of the projection of the displaced point with the present axes, and z
and / be the corresponding variables for the older axes, we have
1+z 1+* '
that IS, z = —j^ z , z = —^ Zi.
Taking now the twelve substitutions in the form of the last set and substi-
tuting, we have a group of tetrahedral non-homogeneous substitutions in the
.form
i Z^^-{l+i) ^V2 + (l+0
^-±A ±^, -\l-i)z+^Jr -\\-i)z-^r
. Z^/2 + (1 - i) . Z^/2 - (1 -0
-\l+i)Z-^2' -\l+i)Z+^f2'
when one of the coordinate planes is a plane through one edge of the
tetrahedron bisecting the opposite edge: each of these gives rise to two
homogeneous substitutions, making 24 in all.
301. The explanations, connected with these groups of substitutions,
implied that certain aggregates of points remain unchanged by the operations
corresponding to the substitutions. These aggregates are (i) the summits of
the tetrahedron, (ii) the summits of the polar tetrahedron — these two sets
together make up the summits of the cube : and (iii) the middle points of the
edges, being also the middle points of the edges of the polar tetrahedron —
this set forms the summits of an octahedron.
When these points are stereographically projected, we obtain aggregates
of points which are unchanged by the substitutions. We therefore project
stereographically with the extremity z of the axis Oz for origin of projection :
and then the projections of x, x, y, y , z, z are 1, — 1, i, — ^, oo , 0, which are
the variables of these points.
Instead of taking factors z—1, z-\-l, ..., we shall take homogeneous
forms z-^ — z^, Z1+Z2, z^ — iz^, z^ + iz^, Z2, ■^i 5 the product of all these factors
equated to zero gives the six points. This product is
t = ZiZo^ (z^* - zi).
1 — 1 1
For the tetrahedron ABCD, the summits A, B, G, D are —r^, — -, —r^;
\/S 1^0 v^
_j^ _i_ __L _J_ J^ A J_ Ji_ zl
V3' V3' V3' V3' V3' V3' \/3' V3' V3 ' ^^^P^^^^^^^^ • ^^^
therefore the variables of the points in the stereographic projection are
^3 _ 1 ' "^ ^' ^3 + 1 ' ' V3 - 1 ' ' V3 + 1 ■
301.] TETRAHEDRAL FUNCTIONS 763
Forming homogeneous factors as before, the product of the four equated to
zero gives the stereographic projections of the four summits of the tetra-
hedron ABGD. This product is
Similarly for the tetrahedron A'B'G'D'; the product of the factors
corresponding to the stereographic projections of its four summits is
^ = ^1^ + 2 V^^i V + Z2*-
And the product of the eight points for the cube is '^, that is,
Tr=5/ + i42iV + .^2'.
All these forms t, , ^ are, by their mode of construction, unchanged
(except as to a constant factor, which is unity in the present case) by the
homogeneous substitutions : and therefore they are invariantive for the group
of 24 linear homogeneous substitutions, derived from the group of 12 non-
homogeneous tetrahedral substitutions. If '^ be taken as a binary quartic,
then is its Hessian and t is its cubicovariant : the invariants are numerical
and not algebraical : and the syzygy which subsists among the system of
concomitants is
a relation easily obtained by reference merely to the expressions for the forms
<|), ^, t.
The object of this investigation is to form Z, the simplest rational
function of z which is unaltered by the group of substitutions. For this
purpose, it will evidently be necessary to form proper quotients of the
foregoing homogeneous forms, of zero dimensions in z-^ and z^. Let R
be any rational function of z, which is unaltered by the tetrahedral
substitutions. These substitutions give a series of values of z, for which
Z has only one value : hence R and Z, being both functions of z and
therefore of one another, are such that to a value of Z there is only one
value of R, so that ii is a rational function of ^.
In particular, the relation between R and Z may be lineo-linear : thus Z
is determinate except as to linear transformations. This unessential indeterm-
inateness can be removed, by assigning three particular conditions to
determine the three constants of the linear transformation.
The number of substitutions in the ^-group is 12. As there will thus
be a group of 12 ^r-points interchanged by the substitutions, the simplest
rational function of Z will be of the 12th degree in z, and therefore the
numerator and the denominator of the fraction for Z, in their homogeneous
forms, are of the 12th degree. The conditions assigned will be
(i) Z must vanish at the summits of the given tetrahedron :
(ii) Z must be infinite at the summits of the polar tetrahedron :
(iii) Z must be unity at the middle points of the sides.
764
TETRAHEDRAL
[301.
As ^ is a fractional function with its numerator and its denominator each
of the 12th degree and composed of the functions ^, '^, t, it must, with
the foregoing conditions, be given by
z =
<|)3
By means of the syzygy, we have
Z:^- 1 : 1 = ^3 : - Us/^St' : ^^
which is Klein's result. Removing the homogeneous variables, we have
Z: Z-1 : l={z'- 2\/^2^ + If : - l2^/^z' (^ - 1)^ : (^ + 2 V^^^ + 1)^ ;
and then ^ is a function of z which is unaltered by the group of 12 tetra-
hedral substitutions of p. 761. And every such function is a rational function
of ^.
This is one form of the result, depending upon the first position of the
axes. For the alternate form it is necessary merely to turn the axes through
an angle of ^tt round the 2^-axis, as was done in § 300 to obtain the new
groups. The result is that a function Z, unaltered by the group of 12
substitutions of p. 762, is given by
^ : ^ - 1 : 1 = (^* - 2^/Sz' - 1)=' : - 12^/Sz' (^ + 1)^ : (z^ + 2 VS^^ _ ly^
It still is of importance to mark out the partition of the plane corre-
sponding to the groups, in the same manner as was done in the case of the
infinite groups in the preceding chapter. This partition of the plane is the
stereographic projection of the partition of the sphere, a partition effected by
the planes of symmetry of the tetrahedron. Some idea of the division may
be gathered from the accompanpng figure, which is merely a projection on
the circumscribing sphere from the centre of the cube. The great circles
Fig. 120.
301.] FUNCTIONS 765
meet by threes in the summits of the tetrahedron and its polar, being the
sections by the three planes of symmetry, which pass through every such
summit, and the circles are equally inclined to one another there : they meet
by twos in the middle points of the edges and they are equally inclined to
one another there. They divide the sphere into 24 triangles, each of which
has for angles ^-tt, ^tt, Jtt. (See case II., § 278.)
The corresponding division of the plane is the stereographic projection of
this divided surface. Taking A as the pole of projection, which is projected
Fig. 121.
to infinity, then A' is the origin : the three great circles through A' become
three straight lines equally inclined to one another; the other three great
circles become three circles with their centres on the three lines concurrent
in the origin. The accompanying figure shews the projection: the points in
the plane have the same letters as the points on the sphere of which they
are the projections : and the plane is thus divided into 24 parts. There are,
in explicit form, only 12 non-homogeneous substitutions: but each of these
has been proved to imply two homogeneous substitutions, so that we have
the division of the plane corresponding to the 24 substitutions in the group.
The fundamental polygon of reference is a triangle such as GA'x.
302. It now remains to construct the function for the dihedral group.
The sets of points to be considered are : —
(i) the angular points of the polygon : in the stereographic projection,
these are
2Trsi
e '^ , for s=0,l, ...,n-l;
766 DIHEDEAL FUNCTION [302.
(ii) the middle points of the sides : in the stereographic projection,
these are
e "■ , for 5 = 0, 1, . . . , w - 1 ; and
(iii) the poles • of the equator which ' are unaltered by each of the
rotations : in the stereographic projection, these are and oo ,
Forming the homogeneous products, as for the tetrahedron, we have, for (i),
for(ii), V = 2,^+z,^;
and, for (iii), W = z^z^ \
these functions being connected by a relation
Because the dihedral group contains 2n non-homogeneous substitutions,
the rational function of z, say Z, must, in its initial fractional form, be of
degree 2n in both numerator and denominator ; and it must be constructed
from U, V, W.
The function Z becomes fully determinate, if we assign to it the following
conditions :
(i) Z must vanish at points corresponding to the summits of the
polygon,
(ii) Z must be infinite at points corresponding to the poles of the
equator,
(iii) Z must be unity at points corresponding to the middle points of
the edges :
and then we find
Z:Z-1.1 = {i(^-- l)r- : {i(^'^ + l)p :-z-,
which gives the simplest rational function of z that is unaltered by the
substitutions of the dihedral group.
The discussion of the polyhedral functions will not be carried further here : sufl&cient
illustration has been provided as an introduction to the theory which, in its various
bearings, is expounded in Klein's suggestive treatise already quoted.
JSa;. 1. Shew that the anharmonic group of § 298 is substantially the dihedral group
for 71=3 ; and, by changing the axes, complete the identification. (Klein.)
Ex. 2. An octahedron is referred to its diagonals as axes of reference, and a partition
of the surface of the sphere is made with reference to planes of symmetry and the axes of
rotations whereby the figure is made to coincide with itself.
Shew that the number of these rotations is 24, that the sphere is divided into 48
triangles, that the non-homogeneous substitutions which transform into one another the
partitions of the plane obtained from a stereographic projection are
' s' Z-1 2+1' Z + t' Z-%^
where ^=0, 1, 2, 3 ; and that the corresponding octahedral function is
Z: Z-\ : 1 = (28 -I- 142*+!)^ : (^^^ _ 3328 _ 33^+1)2 . 1082* (s^ _ i )4. (Klein.)
303.] ELLIPTIC MODULAR-FUNCTIONS 767
303. We now pass from groups that are finite in number to the
consideration of functions connected with groups that are infinite in
number. The best known illustration is that of the elliptic modular-
functions; one example is the form of the modulus in an elliptic integral
as a function of the ratio of the periods of the integral. The general
definition of a modular-function* is that it is a uniform function such that
an algebraical equation subsists between i/r i ^j and ■^{w), where a, j3,
7, S are integers subject to the relation aS-/37=l. The simplest case is
that in which the two functions t^ are equal.
The elliptic quarter-periods K and iK' are defined by the integrals
^K=\ [z(l-z){l-k-'z)]-^dz=Wz{l-z){l-cz)\-^'dz,
.0 Jo
2K'= Wz{\-z){l-k''z)]-^dz= W2{l-z){\-c'z)]-idz,
Jo Jo
where c + c' =1. The ordinary theory of elliptic functions gives the equation
dc dc 4cc"
whatever be the value of c. To consider the nature of these quantities as
functions of c, we note that c = 1 is an infinity of K and an ordinary point of
K', and that similarly c = is an infinity of K' and an ordinary point of K :
and these are all the singular points in the finite part of the plane. The
value c = 00 must also be considered. All other values of c are ordinary
points for K and K'.
For values of c, such that | c j < 1, we have
so that, in the vicinity of the origin,
d fK'^
dc\K 4>K'cc'
= i+o"*" positive integral powers of cL
Hence in the vicinity of the origin
— =--Iogc-HP(c),
where P (c) is a uniform series converging for sufficiently small values of \c\:
and therefore, still in the vicinity of the origin,
^' = --logc + irP(c).
TT
* This is the definition of a modular- function which is adopted by Hermite, Dedekind, Klein,
Weber, and others.
768 ELLIPTIC [303.
Now let the modulus c describe a contour round the origin and return to
its original value. Then K is unchanged, for the c-origin is not a singularity
oi K.
The new value of K' is evidently
--(27ri + \ogc)+KP(c),
IT
that is, iK' changes into 2K + iK'. Hence, ^uhen c describes positively a
small contour round the origin, the quarter-periods K and iK' become K and
2K + iK' respectively.
In the same way from the equation
„, dK „ dK' _ TT
and from the expansion of ^' in powers of c' when \c'\< 1, we infer that
when c' describes positively a small contour round its origin, that is, when c
describes positively a small contour round the point c = 1, then iK' is unchanged
and K changes to K— 2iK'.
It thus appears that the quantities K and iK', regarded as functions of
the elliptic modulus c, are subject to the linear transformations
U{K) = K ) V{K) = K-2iK']
U (iK') =2K + iK'] ' V (iK') = iK'\
without change of the quantity c ; and the application of either substitution
is equivalent to making c describe a closed circuit round one or other of the
critical points in the finite part of the plane, the description being positive if
the direct substitution be applied and negative if the inverse be applied.
When these substitutions are applied any number of times — the index
being the same and composed in the same way for K as for iK' — then,
denoting the composite substitution by P, we have results of the form
PK=ZK + r^iK'
PiK' = ^K + aiK'
where /3, and 7 are even integers, a and S are odd integers of the forms
1 + 4
UTw 2 + Tw l-2w
so that V is compounded of T and U. Hence the substitutions for cc',
regarded as a modular-function, are the infinite group which is derived from
the fundamental substitutions
Uw=w + 2, Tw= .
w
Denoting the modular-function cc' by ^ {w), we have
CC' = % (w) = % (W + 2) = ;^; ( - - )
V wj
To obtain the change in w caused by changing c into c/c, we use the
differential expression
When the variable is transformed by the equation* (1 — y)(l — k^x) = 1— x,
where kfH^ = — k^, the expression becomes
k' [x{l-x){\- k''x)]~^ dx.
When y describes the straight line from to 1 continuously, x also
describes the straight line from to 1 continuously. Integrating between
these limits, we have
A = k'K,
where A is a quarter-period. When y describes the straight line from
to 1/^ continuously, x describes the straight line from to oo continuously,
or, say, the line from to Ijk"^ and the line from Ijk^ to oo continuously.
Integrating between these limits, we have
A -h ^A' = k' (K + iK') + \k' T [x{l- x){l- k-'x)]'^ dx
= k'{K + iK') + k'K,
on using the transformation k^xu = 1 and taking account of the path described
by the variable u : and therefore
iA' = k' {K + iK').
Hence the change of modulus from k to ikfk', which changes c to — cjc, gives
the changes of quarter-periods in the form
A = k'K, iM = k'{K+ iK') ;
and therefore the new value of w, say Wi, is
Wi = w + 1 = 8w.
It therefore follows that, when — cjc' is regarded as a modular-function
of the quotient lu of the quarter-periods K and iK', it must be subject to
the substitutions
S(w)=^w+l, U{w) = iu+2, V{tv)=^ _^^^.
* This is the e(j[aation expressing elHptic functions of k'u in terms of elliptic functions of u.
303.] AUTOMORPHIC FUNCTIONS 771
Evidently S^ = U, and U may therefore be omitted ; V and 8 are the
fundamental substitutions of the infinite group of transformations of w,
the argument of the modular-function c/c'.
As a last example, we consider the function
/ =
(c'^c + iy
It is a rational function of cc, and therefore is a modular-function having the
substitutions Tw and Uw. By § 298, it is unaltered when we substitute
c
— -y for c. It has just been proved that this change causes a change of w
into w-f-1, and therefore /, as a modular-function, must be suT^ject to the
substitution
Sw^w -\- 1.
Evidently S^w = w + 2= Uw, so that U is no longer a fundamental substitution
when aS^ is retained. Hence we have the result that J is unaltered, when w is
subjected to the infinite group of substitutions derived from the fundamental
substitutions
Sw = w + 1, Tw = — ,
w
so that we may write
J =
= J{w) = J{w+l) = j{-^.
This is the group of substitutions considered in § 284 : they are of the
form 5^ , where a, /3, 7, 8 are real integers subject to .the single relation
These illustrations, in connection with which the example in § 298 should be consulted,
suffice to put in evidence the existence of modular-functions, that is, functions periodic
for infinite groups of linear substitutions, the coefficients of which are real integers. The
theory has been the subject of many investigations, both in connection with the modular
equations in the transformation of elliptic functions and also as a definite set of functions.
The investigations are due among others to Hermite, Fuchs, Dedekind, Hiu-witz, and
especially to Klein* ; and reference must be made to their memoirs, or to Klein-Fricke's
treatise on elliptic modular-functions, or to Weber's ElUptische Functionen, for an exposi-
tion of the theory.
304. The method just adopted for infinite groups is very special, being
suited only to particular classes of functions : in passing now to linear
substitutions, no longer limited by the condition that their coefficients are
real integers, we shall adopt more general considerations. The chief
purpose of the investigation will be to obtain expressions of functions
characterised by the property of reproduction when their argument is
subjected to any one of the infinite group of substitutions.
* Some references are given in Enneper's Elliptische Functionen, (2*« Aufl.), p. 482.
49—2
772 CONSTRUCTION OF ' [304.
The infinite group is supposed of the nature of that in § 290: the
members of it, being of the form
(ni^:)' - <^'/'<^)>'
are such that a circle, called the fundamental circle, is unaltered by any of the
substitutions. This circle is supposed to have its centre at the origin and
unity for its radius.
The interior of the circle is divided into an infinite number of curvilinear
polygons, congruent by the substitutions of the group : each polygon contains
one, and only one, of the points in the interior associated by the substitutions
with a given point not on the boundary of the polygon. Hence corresponding
to any point within the circle, there is one and only one point within the
fundamental polygon, as there is only one such point in each of the polygons :
of these homologous points the one, which lies in the fundamental polygon
of reference, will be called the irreducible point. It is convenient to speak of
the zero of a function, implying thereby the irreducible zero : and similarly
for the singularities.
The part of the plane, exterior to the fundamental circle, is similarly
divided : and the division can be obtained from that of the -internal area by
inversion with regard to the circumference and the centre of the fundamental
circle. Hence there will be two polygons of reference, one in the part of the
plane within the circle and the other in the part without the circle : and
all terms used for the one can evidently be used for the other. Thus the
irreducible homologue of a point without the circle is in the outer polygon
of reference : for a" substitution transforms a point within an internal polygon
to a point within another internal polygon, and a point within an external
polygon to a point within another external polygon.
Take a point z in the interior of the circle, and round it describe a small
contour (say for convenience a circle) so as not to cross the boundary of the
polygon within which 2 lies : and let Zi be the point given by the substitution
fi{z). Then corresponding to this contour there is, in each of the internal
polygons, a contour which does not cross the boundary of its polygon : and as
the first contour (say Cq) does not occupy the whole of its polygon and as the
congruent contours do not intersect, the sum of the areas of all the contours
Cj is less than the sum of the areas of all the polygons, that is, the sum is
less than the area of the circle and so it is finite.
If fii be the linear magnification at Zi, we have
dzi
f^i
dz
and therefore, if mj be the least value of the magnification for points lying
within Co, we have
Ci>7n/G..
304]
A CONVERGING SERIES
773
The point is the homologue of z = by the substitution
z, ^ VI , and therefore — 84/7^ lies without
Ji2 + OiJ
the circle : though, in the limit of i infinite, it
may approach indefinitely near to the circum-
ference*.
Let this point be G : and through G and
0, the centre of the fundamental circle, draw
straight lines passing through the centre of
the circular contour. Then evidently
, , „ 1
^i= |7i|
GP''
and, if Mi be the greatest magnification, then
Fig. 122.
1
GQ^
so that
Mj^GP"
mi~ GQ''
Now G is certainly not inside the circle, so that GQ is not less than RA :
thus
GP_ PQ^-,,A^ . AB RB
GQ~ '^ GQ ^ GQ^ '^ RA^RA'
which is independent of the point G, that is, of the particular substitution
fi (z). Denoting ( ^^ j by K, we have
M, ^
< K,
or
Evidently fjb^ is finite.
Now
and therefore
so that
mi
Mi < Kmi.
\yiZ +
-s-|2 = /^i < Mi < Krrii
1 K^
\'yiZ + bif t>o
S \yiz + Si\-^ < jT S C'i-
4=0 ^0 r=o
* For, in § 284, when the coefficients are real, a point associated with a given point may, for
i = cx> , approach indefinitely near to a point on the axis of x : and then, by the transformation of
§ 290, we have the result in the text.
774
A CONVERGING SERIES
[304.
It has been seen that 2 Ci is less than the area of the fundamental circle and
is therefore finite : hence the quantity
00
is finite. It therefore follows that 2 /*/ is an absolutely converging series.
i =
00
Similarly, it follows that 2 /uLi"^ is an absolutely converging series for all
i=0
integral values of m that are greater than unity*. This series is evidently
i =
and the absolute convergence is established on the assumption that z lies
within the fundamental circle.
Next, let z lie without the fundamental circle. If z coincide with some
one of the points — Si/l and H{z)
he a rational function of z. The singularities of B are : —
(i), the accidental singidarities of H {z) and the points homologous
with them by the substitutions of the group : all these points are
accidental singularities of B {z) ;
(ii), the points — S,:/7i, which are the 'points homologous with z = co by
the substitutions of the group : all these points, if not ordinary
points of B {z), are accidental singularities ; and
(iii), the essential singularities of the group: these lie on the fundamental
circle and they are essential singularities of B {z).
If H{z) had any essential singularity, then that point and all points homo-
logous with it by substitutions of the group would be essential singularities
of B {z). The function B {z), thus defined, is called* Thetafuchsian by
Poincare.
* Acta Math., i. i, (1882), p. 210.
305.] PSEUDO-AUTOMOEPHIC PROPERTY 777
If the group belong to the first, the second, or the sixth family,
it is known that the circumference of the fundamental circle enters into
the division of the interior of the circle (and also of the space exterior to
the circle) only in so far as it contains the essential singularities of the
group. But if the group belong to any one of the other four families,
then parts of the circumference enter into the division of both spaces.
In the former case, when the group belongs to the set of families,
made up of the first, the second, and the sixth, the circumference of the
fundamental circle is a line over which the series cannot be continued : it
is a natural limit (§ 81) both for a function existing in the interior of the
circle and for a function existing in the exterior of the circle : but neither
function exists for points on the circumference of the fundamental circle.
The series represents one function within the circle and another function
without the circle.
It has been proved that the area outside the fundamental circle can
be derived from the area inside that circle, by inversion with regard to
its circumference. Hence a function of z, existing only outside the funda-
mental circle, can be transformed into a ftmction of — , and therefore also
of - , existing for points only within the circle. When, therefore, a group
belongs to the first, the second, or the sixth family, it is sufficient to consider
only the function defined hy the series for points within the fundamental
circle : it will be called the function @ {z).
In the latter case, when the group belongs to the third, the fourth, the
fifth, or the seventh families, then parts of the circumference enter into the
division of the plane both without and within the circle. Over these parts
the function can be continued : and then the series represents one {and only
one) function in the two parts of the plane : it will be called the function {z).
306. The importance of the function © {z) lies in its pseudo-automorphic
character for the substitutions of the group, as defined by the property now
to be proved that, if ^ he any one of the substitutions of the group, then
«.
Since the whole integral must prove to be a real quantity, we omit the
parts — -x — . log M as in the aggregate constituting an evanescent (imaginary)
quantity : hence we have
m
2^ (- 9a + 96)
as the part corresponding to the side AB. In this expression, (p^ is the angle
required to turn AG into a direction parallel to A'C, and ^j is the angle
required to turn QB, that is, GB into a direction parallel to QB', that is,
G'B', both rotations being taken positively. Thus
(^a = inch A'G' — incl. AG,
6 = 27r - incl. BG + incl. B'G' ;
and therefore
<^„ _ (^j, = - 27r + incl. A'G' - incl. B'G' + incl. BG - incl. AG
= -27r + Ci' + Ci,
where Ci and c/ are the angles AGB, A' G'B' respectively. Hence, if we take
c and c to be the external angles AGB, A' G'B' as in the figure, we have
c + Ci = 27r = c' + c/,
and therefore (fib— 4>a = c + c' — 27r.
The part corresponding to the arc AB in the above integral is therefore
(c + c'-27r).
ztt
There are no sides of the second kind in the path of integration, because the
function is supposed to exist only within the circle. Therefore the whole
excess is given by
£;=^X(c + c'-27r),
the summation ' extending over those sides of the polygon, being in number
half of the sides of the first kind, which are transformed into their conjugates
by the fundamental substitutions of the group.
782
EXCESS OF NUMBER OF ZEROS
[307.
Draw all the pairs of tangents at the extremities of the bounding arcs
of the fundamental polygon of reference :
then the angles, such as c and c' above,
are internal angles of the rectilinear
polygon formed by the straight lines.
The remaining internal angles of this
new polygon are the angles at which
the arcs cut, which are the angles of
the curvilinear polygon : and therefore
their sum is the sum of the angles in
the cycles, that is, the sum is equal to
^27r
iU-i
2-;
Fig. 125.
where ^^^ is the sum of the angles in
one of the cycles. Now let 2n be the number of sides of the first kind in
the curvilinear polygon, so that n is the number of fundamental substitutions
in the group : hence the number of terms in the above summation for E is
n, and therefore
E = -mn + ^t(c + c').
Moreover, the rectilinear polygon has 4n sides : and therefore the sum of the
But &is sum is equal to S (c + c') + 2 —
internal angles is (4?i - 2) tt.
where the first summation extends to the different conjugate pairs and
the second to the different cycles : thus
Therefore
(4w - 2) TT = 2 (c + c') + 2'jrt
E = — mn + m (2?i — 1) — mX
= m{n — l — ^ —
1^
where the summation extends over all the different cycles in the fundamental
polygon. Hence for a function, which is constructed from the additive
element H {z) and exists only within the fundamental circle of the group, the
excess of the number of its irreducible zeros over the number of its irreducible
accidental singularities is
m
72-1-S-
where m is the parametric integer of the function constructed in series, 2n is
the number of sides of the first kind in the fundamental polygon, — is the sum
307.] ^ OVEE NUMBER OF SINGULARITIES 783
of the angles in a cycle of the first kind of corners, and the summation extends
to all these cycles.
The number of irreducible accidental singularities has already been
obtained ; it is finite, and thus the number of irreducible zeros is finite.
Secondly, let the function exist all over the plane : then the irreducible
points are (i) points lying within (or on) the boundary of the fundamental
polygon of reference within the fundamental circle and (ii) points lying
within (or on) the boundary of the fundamental polygon of reference without
the fundamental circle, the outer polygon being the inverse of the inner poly-
gon with regard to the centre. For such a function the excess of the number
of irreducible zeros over the number of irreducible accidental singularities is
the integral
1 [(d'{z)
27ri J S (s) ^^'
taken positively round the boundaries of both polygons. We shall assume
that there are no zeros and no infinities on the path of integration ; the
result can, however, be shewn to be valid in the contrary case.
For the sides of the internal polygon that are of the first kind the value
of the integral is, as before, equal to
7n[n — l — z —
and for the sides of the external polygon that are of the first kind, the value
is also
1
m ( n — 1 — S
V /",
Let the value of the integral along the sides of the second kind in
the internal polygon be /. Those lines are also sides of the second kind
in the external polygon ; but they are described in the sense opposite to
that for the internal polygon, the integral being always taken positively:
hence the value of the integral along the sides of the second kind in the
external polygon is — /.
Hence the excess of the number of irreducible zeros over the number of
irreducible accidental singularities of a function % {z), which is constructed
from the additive element E {z) and exists all over the plane, is
2m {n—l—'2 —
where the summation extends over all the cycles of the first category of either
{but not both) of the fundamental polygons of reference.
As before, the number of irreducible zeros of such a function is finite,
because the number of irreducible accidental singularities is finite.
784 FUCHSIAN FUNCTIONS [307.
In every case, this excess depends only upon
(i) the parametric integer m, used in the construction of the series :
(ii) the number of sides, 2w, of the first kind in the polygon of
reference :
(iii) the sum of the angles in the cycles of the first category.
Ex. Prove that a corner belonging to a cycle of the first category is in general a zero '
of order ^, such that
p= -m (mod. fj),
where 27r//i is the sum of the angles in the cycle : and discuss the nature of the corners
which belong to cycles of the remaining categories. (Poincare.)
308. We are now in a position to construct automorphic functions, using
as subsidiary elements the pseudo-automorphic functions which have just
been considered.
For, if we take a couple of these functions, @i and @2j associated with a
given infinite group, characterised by the same integer m, and arising through
different additive elements H {z), then we have
\'yz + h
= (7^ + sr«@,(4
where ^ is any one of the substitutions of the group ; and therefore
\^z + h) @i(^)
^ (az + /3\ @2 {2)
&f) = ^»(^)'
\^z + S /
that is, the quotient of two such functions is automorphic. Denoting the
quotient by Pn(z)*, we have
^cizj\- /3^
^yz
the automorphic property being possessed for each of the substitutions.
It thus appears that such functions exist : their essential property is
that of being reproduced when the independent variable is subjected to any
of the linear substitutions of the infinite group.
The foregoing is of course the simplest case, adduced at once to indicate
the existence of the functions. The construction can evidently be general-
ised: for, if we have any number of functions @i, ..., ©^, ^i, ..., ^g with
characteristic integers wij, ..., mr,n^, ..., ng and all associated with one group
* Poincare calls such functions Fuchsian functions: as already indicated (§ 297), I have
preferred to associate the general name automorphic with them. But, because Poincare himself
has constructed one class of such functions by means of series as in the foregoing manner, his
name, if any, should be associated with this class : the symbol Pn {z) is therefore used.
308.] TWO CLASSES OF AUTOMORPHIC FUNCTIONS 785
while constructed from different additive elementary functions H (z), then,
denoting
^i(^) ^s{^)
by Pn (2), we evidently have
SO that, provided only 2 ti^ ^ 2 Wg,
3=1 9=1
the function is automorphic. If we agree to call m, the integer characteristic
of a pseudo-automorphic function, the degree of that function, then the quotient
of two products of pseudo-automorphic functions is autovnorphic, provided the
products be of the same degree.
There are evidently two classes of automorphic functions: those which
exist all over the plane, and those which exist only within the fundamental
circle. The classes are discriminated according to the composition of the
functions from the subsidiary pseudo-automorphic functions.
When the pseudo-automorphic functions, which enter into the composi-
tion of the function, exist all over the plane, then the automorphic function
exists all over the plane. But when the pseudo-automorphic functions, which
enter into the composition of the function, exist only within the fundamental
circle, then the automorphic function exists only within the circle.
Ex. Consider the quantities go, and ^3, defined in § 123. We have
o, = 6022 -. TT ,
where the double summation extends over all positive and negative integer values of TOi
and wi2, simultaneous zeros alone excepted. Writing w = a)]/(»2, we have
and therefore
1
Oo (c<)) = 60a)2 22 7 ; TT ;
where
yyod-
Mi = mia + m^y, M^ = m-^j^ + m^^.
Because ab — ^y=\, we have
m^ = Mxb - M^y, »i2 = - Mi^ + M^a.
If then h, iS, y, 8 be integers, subject solely to the condition aS-/3y = l, it follows that, to
every pair of integer values of mj and 7^2, there corresponds one pair (and only one pair)
of integer values of i/j and M^ ; and conversely. Also, simultaneous zeros for the one set
are simultaneous zeros for the other ; so that
1 1
22 .,. . ,,,, =22
{Mia + M^y {m^ca + mzY'
50
786 ESSENTIAL SINGULAEITIES OF AUTOMORPHIC FUNCTIONS [308.
Consequently,
Similarly, as
we have
qr, (a,) = 14022 , rs
= 140w2-<'22 -. ; r^,
It therefore follows that g'^gi~^ is automorphic ; and if we take
J:J-\:\=gi: 27^3^ : gi - ^^gi ( = A),
then J is an automorphic function, such that
where a, /3, y, S are integers subject to the condition ah-i^y=\. Evidently ^2 (<») ^-iid
^3 (o)) are pseudo-automorphic for the group. Taking
ei=/i(l + c'), e3=-/x(l+o), e2=M(c-c')j
where c + c' = l, we have
5r2=-12^2(cc'-l), 5f3= -4/i3 (2 + cc')(c-c'),
A=16.272.,i6c2c'2^
so that
, (1-C + C2)3
^2^ C^(l-C)2 '
as in § 303, where other examples are given.
309. It is evident that all the essential singularities of an automorphic
function, thus constructed, lie on the fundamental circle. For whether the
pseudo-automorphic functions exist only within that circle or over the whole
plane, all their essential singularities lie on the circumference : so that,
whatever be the constitution of the various subsidiary pseudo-automorphic
functions, all the essential singularities of the automorphic function lie on
the fundamental circle.
Next, the number of irreducible zeros of an automorphic function is equal
to the number of its irreducible accidental singularities. For an irreducible
zero of an automorphic function is either (i) an irreducible zero of a factor
in the numerator or (ii) an irreducible accidental singularity of a factor in
the denominator ; and similarly with the irreducible accidental singularities
of the function. The numerator and the denominator may have common
zeros ; this will not affect the result.
First, let the automorphic function exist only within the circle : then
each of its factors exists only within the circle. The space without the circle
is not significant for any of the factors of the function, because they do not
309.] LEVEL POINTS OF AUTOMORPHIC FUNCTIONS 787
there exist. Let ei, ..., e^, e/, ..., e/ be the excesses of zeros over accidental
singularities for the pseudo-automorphic functions within the fundamental
circle : then
eg = 7W„ ( n — 1 - 2 —
\ /til
where n and S — are the same for all these functions, and
Now the excess of zeros over poles in the denominator becomes, after the
above explanation, an excess of poles over zeros for the automorphic
function : hence, for this automorphic function, the excess of zeros over
accidental singularities is
r s
3=1 g=l
= f /?, — 1 — X — ) ( S rria— X ric
= 0,
1
H^ \q = l g = l
by the condition X mq= 2 iiq. Hence the number of irreducible zeros of
9=1 q=l
the automorphic function is equal to the number of irreducible accidental
singularities.
Secondly, let the automorphic function exist all over the plane ; then
all its factors exist all over the plane. For the present purpose, the sole
analytical difference from the preceding case is that each of the quantities e
now has double its former value : and therefore the excess of the number of
zeros over the number of poles is
2{n—l—2 — ){ Sm^— 2 Ug
\ fliJ \q = l 3 = 1
which, as before, vanishes. Hence the number of irreducible zeros of the
automorphic function is equal to the number of its irreducible accidental
singularities.
It follows, as an immediate Corollary, that the number of irreducible
points for which an automorphic function assumes a given value is equal to
the number of its irreducible accidental singularities. For
Pn{z)-A,
where J^ is a constant, is an automorphic function: the number of its
irreducible accidental singularities is equal to the number of its irreducible
zeros, that is, it is equal to the number of irreducible points for which
Pn {z) assumes an assigned value.
50—2
788 DIFFERENT FUNCTIONS FOR ONE GROUP [309.
Moreover, each of these numbers is finite : for the number of irreducible
zeros and the number of irreducible accidental singularities of each of the
component pseudo-automorphic factors is finite, and there is only a finite
number of these factors in the automorphic function. The integer, which
represents each number, will evidently be as characteristic of these functions
as the corresponding integer was of functions with linear additive periodicity.
Note. The preceding method, due to Poincar6, of expressing the pseudo-
automorphic functions as converging infinite series of functions of the
variable, is not the only method of obtaining such functions. It was
shewn that uniform analytical functions can be represented either as
converging series of powers or as converging series of functions or as
converging products of primary factors, not to mention the (less useful)
forms intermediate between series and products. The representation of
automorphic functions as infinite products of primary factors is considered
in the memoirs of Von Mangoldt and Stahl, already referred to in | 297.
310. Let Pni{z), Pn2{z), say Pj and P^, be two automorphic functions
with the same group, constructed with the most general additive elements ;
and let the number of irreducible zeros of the former be k^, and of the
latter be k^,.
Then for an assigned value of P^ there are /Ci irreducible points : P^ has
a single value for each of these points, and therefore it has Kj values alto-
gether for all the points, that is, it has k^ values for each value of Pj.
Similarly, Pj has k.2 values for each value of Pg. Hence there is an alge-
braical relation between Pj and Pg of degree k^ in P^ and of degree k^ in Pg,
which may be expressed in the form
Let Pn{z), say P, be any other uniform automorphic function, having
the same group as Pj and P^', and let k be the number of its irreducible
zeros. Then we have an algebraical equation
F,iP,P,) = Q,
which is of degree Ki in P and of degree /c in Pj ; and another equation
P,(P,P,) = 0,
which is of degree «2 in P and of degree ic in Pg. The last two equations
coexist, in virtue of the relation
P,,(A,P,) = o
satisfied by P^ and Pg. Since Pi = = Po coexist, the ordinary theory of
elimination leads to the result that the uniform function P can be expressed
rationally in terms of Pj and Pg, so that we have the theorem that evei'y
automorphic function associated with a given group can be expressed rationally
in terms of two general a. utomorphic functions associated with that gi^oup: and
between these two functions there exists an irreducible algebraical relation.
310.] ALGEBRAICAL RELATIONS 789
The genus (§ 178) of this algebraical relation can be obtained as follows.
Let N denote the genus of the group, determined as in § 293 : then the
fundamental polygon of reference, if functions exist only within the circle, or
the two fundamental polygons of reference, if functions exist over the whole
plane, can be transformed into a surface of multiple connectivity 2iV+ 1. The
automorphic functions are uniform functions of position on this surface; and
hence, as in Riemann's theory of functions, the algebraical relation between
two general uniform functions of position, that is, between two general auto-
morphic functions is of genus N, where N is the genus of the group*.
It is now evident that the existence-theorem and the whole of Riemann's
theory of functions can be applied to the present class of functions, whether
actually automorphic or only pseudo-automorphic. There will be functions
of the same kinds as on a Riemann's surface : the periods will be linear
numerical multiples of constant quantities acquired by a function when its
argument moves from any position to a homologous position or returns to its
initial position. There will be functions everywhere finite on the surface,
that is, finite for all values of the variable z except those which coincide with
the essential singularities of the group. The number of such functions,
linearly independent of one anot-her, is N ; and every such function, finite
for all values of z except at the essential singularities, can be expressed as
a linear function of these N functions with constant coefficients and (possibly)
an additive constant. And so on, for other classes of functions f.
311. Because Pn {z) is an automorphic function, we have
\'yz + h'
and therefore, as aS — ^^7 = 1,
Hence, if @ {z) be a pseudo-automorphic function with m for its characteristic
integer, so that
®(|Tf) = ^^^ + ^)"®<^>-
0iZ-\- 13\
@
r^z + hj B {z)
p^./az + lBW^ {Pn(z)Y
have
J P-n'
\yz + B/
* It may happen that, just as in the general theory of algebraical functions, the genus of the
equation between two particular automorphic functions may be less than N : thus one might be
expressed rationally in terms of the other. The theorems are true for functions constructed in
the most general manner possible.
t The memoirs by Burnside, quoted in § 297, develop this theory in full detail for the group
which has its (combined) polygons of reference bounded by 2?i circles with their centres on the
axis of real quantities, the group being such that the pseudo-automorphic functions exist over
the whole plane.
790
DERIVATIVES OF AUTOMORPHIC FUNCTIONS
[311.
that is, © (2) [Pn {z)]~^ is an automorphic function. Such a function can
be expressed rationally in terms of Pn{z) and some other function, say of
P and Q : hence the general type of a pseudo-automorphic function with
a characteristic integer m is
(f r^(^' «■
where / is a rational function.
Corollary. Two automorphic functions P and Q, belonging to the same
group, are connected by the equation
For evidently unity is the characteristic integer of the first derivative of an
automorphic function.
This equation can be changed to
§=fiP. Q).
where / is a rational function : moreover, P and Q are connected by an
equation
FiP,Q) = 0,
which is an algebraical rational equation, and can evidently be regarded as
an integral of the above differential equation of the first order, all trace of
the variable 2 having disappeared. Evidently the form of/ is given by
dF j.,y~, r\\^F ^
Again, denoting ^ by ^, and Pn ( — — -» j by 11 (^), we have
say
Then
{yz + Sf
27
jyz +
P"
I'^P'
so that
n'
rr
= (yz + 8y
= 2y{rYZ + 8) + {jz + 8y-^
P"
rP'
2Y + 2y{yz + 8)^+(yz + 8Y(^~
P'2
and therefore ^[7 ~ f \ fF [ ={yz + oy
P' MP'
whence
n'2
P'2
where [P, z] is the Schwarzian derivative. It thus appears that, if P be an
311.] DIFFERENTIAL EQUATIONS 791
automorphic fiinction, then (P, 2] P'~^ is a function automorphic for the
same group.
But between two automorphic functions of the same group, there subsists
an algebraical equation : hence there is an algebraical equation between
P and [P, 2] P'~^, that is, P {2), an automorphic function of 2, satisfies
a differential equation of the third order, the degree of which is the integer
representing the number of irreducible 2eros of P and the coefficients of which,
where they are not derivatives of P, are functions of P only and not of the
independent variable.
This equation can be differently regarded. Take
2/, = P'i y, = 2P'^;
then it is easy to prove that
ld'y,_ld^y,_^{P,2]
y^dP^ y^dP^ 2 p/^ •
The last fraction has just been proved to be an automorphic function of 2 ;
and therefore it is rationally expressible in terms of P and any other general
function, say Q, automorphic for the group. Then y^ and y^ are independent
integrals of the equation
% = y^{P,Q),
where Q and P are connected by the algebraical equation
F{P,Q) = 0.
. Conversely, the quotient of two independent integrals of the equation
^, = y(P,Q),
where Q and P are connected by the algebraical equation
P(P, Q) = 0,
can be taken as an argument of which P and Q are automorphic functions :
the genus of the equation P = is the genus of the infinite group of sub-
stitutions for which P and Q are automorphic*.
Ex. One of the simplest set of examples of automorphic functions is furnished by
the class of homoperiodic functions (§ 116). Another set of such examples arises in the
triangular functions, discussed in § 275 ; they are automorphic for an infinite group, and
the triangles have a circle for their natural limit. A third set is furnished by the polyhedral
functions (§§ 276—279).
As a last set of examples, we may consider the modular-functions which were obtained
by a special method in § 303.
* Klein remarks (Math. Ann., t. xix, p. 143, note 4) that the idea of uniform automorphic
functions occurs in a posthumous fragment by Eiemann (Ges. Werke, number xxv, pp. 413 — 416).
It may also be pointed out that the association of such functions with the linear differential
equation of the second order is indicated by Riemann.
792 MODULAR-FUNCTIONS [311.
First, we consider them in illustration of the algebraical relations between functions
automorphic for the same group. It follows, from the construction of the group and the
relation of c to w, that, in the division of the plane by the group with Uw and Vw for its
fundamental substitutions, where
^ ^ w
uw = w + 2, Vw=z — jr— ,
1— 2w
there is only a single point in each of the regions for which c has an assigned value ; hence,
regarding c as an automorphic function of w, the number k (§ 310) is unity. If there be
any other function C of w, automorphic for this group, then between C and c there is an
algebraical relation of degree in C equal to the number k. for c, that is, of the first degree
in G. Hence every function automorphic for the group, ivhose fundamental substitutions
are U and V, where
w
Uw = W + '2, Vw=- ;r— ,
' 1 - 2w
is a rational function of c.
In the same way, it can be inferred that every function automorphic for the group,
whose fundamental sicbstitutions are
Uw=w + 2, Tw= ,
is a rational function of cc' ; and that every function automorphic for the group, whose
fundamental substitutions are
Sw = iv-\-l, Tw= ,
w
that is, automorphic for all substitutions of the form -^, where a, b, c, d are real
integers, such that ad-bc=\, is a rational function of J—^-^j v^ •
Secondly, in illustration of the general theorem relating to the differential equation
of the third order which is characteristic of an automorphic functioUj we consider the
iK'
quantity c as a function of the quotient of the quarter-periods. Let z denote -^ : then
because every function, automorphic for the same group of substitutions as c, is a rational
function of c, we have
{c z\ .
^-^ = I'ational function of c:
* c^
and therefore, by a property of the Schwarzian derivative,
{z, c]= —same rational function of e.
By known formulae of elliptic functions, it is easy to shew that
thus verifying the general result.
Similarly, it follows that j-^ > ^j-, where O — cc', is a rational function of cc', the actual
value being given by
UK' J 1 - 5(9 -h 16^2
K ' \ 2^2(1 _ 4^)2 '
and that j-^j J\ is a rational function of J, the actual value being given by
\iK' 1 _ 16J^-mJ-330
In this connection a memoir by Hurwitz* may be consulted.
* Math. Ann., t. xxxiii, (1889), pp. 345—352.
311.] CONCLUSION 793
The preceding application to differential equations is only one instance
in the general theory which connects automorphic functions with linear
differential equations having algebraic coefficients. This development
belongs to the theory of differential equations rather than to the general
theory of functions : its exposition must be reserved for another place.
Here my present task comes to an end. The range of the theory of
functions is vast, its ramifications are many, its development seems illimit-
able : an idea of its freshness and its magnitude can be acquired by noting
the results, and appreciating the suggestions, contained in the memoirs of
the mathematicians who are quoted in the preceding pages.
MISCELLANEOUS EXAMPLES.
I.
1 . Find the curves which cut orthogonally the family of curves r^ r^ r^ = a, where ri , r2 , r^
are the distances of a variable point from three fixed points.
2. P and Q are conjugate functions, so that P + ^■$ is a function f{z) of a compl^;^
variable z=x-\-iy and
P-§=(cos.r + sin^-e-'')/(2cos.r-e!'-e-!').
Find /(?), subject to the condition /(i?r) = 0.
3. Evaluate the integral
j _„l-2a;cos^ + ^2 ■*'
where a, 5 are real, obtaining the necessary limitation of the values of a, for which the
integral is finite.
4. Evaluate rigorously the integrals
n sinao?.^
^ ' j _i 1 — 2«cosa + .»2'
(2) dx,
Jo ^
in which a may have any real value.
State in each case the value or values of a for which the integrals are discontinuous
functions of a.
5. Prove that, if a and b are real, then
f °° sin (x—a) ,
I ^ -dx=ir,
J _oo X — a
/■* sin {x - a) sin (^ -b) j _ sin (a-b)
J -oc {x-a){x-h) _ {a-b) '
6. Prove that, if a is positive,
jo 2Va/
Justify differentiating the integral under the sign of integration any number of times
with respect to a, and so obtain the formula
/"" , , , 1.3 (2%-l) f TT 1^
Finally justify the deduction of the formula
1 6*
I e-aa;'^cos26^'c?^=-i— i e «,
by expansion of the cosine as a power series and integration term by term.
MISCELLANEOUS EXAMPLES 796
7. Prove that, if < w < 1,
/;
smo; ,
^" 2r(w)sin(i7r?i)
8. Prove that
X cos ax , TT^e-aT
ax— ■
sinh a; (1 + e-ajr)2 '
-=loar2.
{l-hx'^) cosh ^TT^
9. Obtain expansions of the cosine and sine integrals defined by
... f^cosx J ... f'^sinx ^
ci[x)= j — —ax, si{x)= I dx,
J X ^ J ^ ^
appropriate for calculation (i) when x is large, and (ii) when x is small.
/■" g—iex
Express I ,„ „ dk in terms of the cosine integral and sine integral.
10. Shew that, if m is real and positive,
sin mx
/:
11. Prove that, when a is real and positive,
°° cos a? , 7re~«,
12. By contour integration (or any method) prove that
* sin a? , TT , . ,
_.^(^2-2.r+2)^^ = 2^^^-"°'^ + '^^y)'
1 — cos X
/;
/;
j _oo « {x"^ - "ix + i) """~2e
the angle y being the unit of circular measure.
13. Prove that, when n is even,
, r sinhM^ "I '•='*, , r 2rw . , 1
tan-M — T-^ 7^^r= 2 tan-Msec;r^ — vsmha?}-.
(cosh(n + l)^J ,.=1 ( 2»+l j
14. Prove that, if a > 0, the value of the integral
/
taken round three sides of a rectangle from
z = R to R + ni, R + ni to —R + ni, —R + ni to —R,
will tend to zero as R and n tend to oc (n being an even integer).
Deduce, or otherwise prove, that
/;
l+x'
15. Prove that, if a < 1,
tanh (^ ttx) 5 dx = a cosh a — sinh a log (2 sinh a).
sinrx , tt ( ,, 1
a.r = - i coth Trr -
e=^-l 2 (. Trr
1 f^ -x'^-e^ _ 1 /■"
=lH 1 v.
796 MISCELLANEOUS EXAMPLES
16. Prove that, if a is positive,
X — sin X
/:
(^^=|7ra-*(^a2-a + l-e-«)
x^{cfi + x^)
If ■
.■?; = a+^sin ^,
obtain the expansion of in powers of t, in the form
x — a
cos a . =° f^ o?«' + i(sin"a)
- 2 ^ -
x — a t sin a sin a n=\ (?i + 1) ! *i c?a'
18. By evaluating the integral (where c is real)
/
— cosh {v/(c2 - z^)} &,
taken round a path which consists of the real axis and a large semicircle, prove that
/
Prove similarly that
cosh {x/(c2 - x'^)} dx = \'n (cosh c - ^).
r cos iax) ''°Mf "' .f ^^ dx = Uh {c v/(l -«^)},
if 0 1.
19. Criticise the following argument : — ~
"Putting 2=rcos^ = rja; r^^x^^-y^-^-z^, ar = rsin^, the functions (where % is a
positive integer)
2^i
2»
(2 + ^ar cos (^)" 0?^',
277
(2 + m COS 0) - ("^ + 1) C?^
are finite over a unit sphere, and for ar = they reduce to r" and r~^^'^^^ respectively.
They must therefore be equivalent to r»P„ {]i) and r-('' + i)P^(ju.) respectively."
Shew that the conclusion is invalid with regard to the second integral when z is
negative : and give the correct result.
Prove that, if the coefficients a^ are such that the series 2 |a„ — a^+i | is convergent,
then the series 2a„ P„ (/x) converges uniformly in any interval for /x which falls within the
interval ( — 1, +1).
Determine an asymptotic formula for the sum of 2 P^ {fj), as r tends to 1 and fi
tends to 1.
20. Prove that, when the real part of ?i is greater than — 1,
l fc + cci , ., ...
where J^ (x) is Bessel's Function of order n satisfying the equation
Establish the relations
/;
/
dx^ X dx \ x^J "
bn ''
e~ax"'Jn{bx)x^'-^dx = j^^^^e 4«, ?i>-l.
1 (26')"
e-axJn(bx)x'^dx=-= T(n + h) — ^ — - — rr? «>-i-
MISCELLANEOUS EXAMPLES 797
21. Prove that e«»'cose may be expanded in either of the two forms
JoW + 2 2 ^■»J„(r)cos?^<9,
2 {2n + l) i^Jn^i (r) P„ (cos 6).
By employing the first expression, or otherwise, prove the theorem
Jo{(?-H/2-2rr'eos(9)4} = Jo(0«4(''') + 2 2 J"„ (r) /„ (/) cos to^.
22. Prove that
^m /•"& ri
'^™^^) = 2;;?j./^Pl2
where the path of integration encloses the negative real axis in the w-plane.
Prove, (i), that
/:
J^,{qx)e-^^^-x"^^^dx=~'^— - ^^
(ii), that, with In{x) — i~'^J^{ix\
1 r**
-— . ey'h{z)z'^dz
_ 8m{n + m)TT T{n+m + l) /n + ??i + l n + 7n + 2 ^A
- ~ 2»r(?i+l)2/™+'" + i V 2 ' 2 ''* + ^'.y y'
where |y | > 1, and the real parts of ay, 6y, are negative ; and
(iii), that j ^ J^{px)J„{qx)a;r--^^dx=^ 2--^-4{n-my
where q> p, n'>m> -1.
23. Shew that
eXcose = 2«-in(«-l)^r(n + X-) C,»(cos^)%^\
fc=o '>■ *
where Cfc" (cos 5) is the coefl&cient of k'' in the expansion of ( 1 — 2A cos ^ 4- A^) ""■ in powers of h.
Prove further that
27^^ j (._ooi \
^(x6\ (if
TyT'
where c is any real positive quantity, the real part of n is greater than - 1 and the real
parts of (a ± b)^ are positive, and hence deduce the addition theorem for
J„J{a^ + b^-2abGosd) ,
{J(a^^b^'-2ab cos 6)}'' '
/■" e^f*"^*^) , . 1 , 27r >_i ■
24. If X, g, be positive numbers, prove that / . dz is equal to jY?x^) ^^
to 0, according as c is positive or negative. Explain how this integral may be employed
in the transformation of a multiple integral taken through a limited domain.
Hence or otherwise, prove that
where the double integration is taken for all values of ^, ?;, such that 1 — 2~p^^^^ ^^^
6q is the positive root of the equation 1 — ^-y-^ - tAxa - -^ = 0.
798 MISCELLANEOUS EXAMPLES
25. If the contour of the integral separates positive and negative sequences of poles
of the subject of integration and at a large distance from the origin is parallel to the
imaginary axis, shew that, for general 'complex values of a^, Cg, ^i and /32, the integral
^-. / r (ai + s) r {a2+s) r (/3i -s) r (/^2-«) <^«
27
is convergent and equal to
r (ai + ^i) r (g^ + 132) r (ai + jgg) r (a2 + ^l) '
r(ai + a2+^i+^2)
Deduce, or obtain independently, Riemann's transformation
F{2a,2^; y; x} = F{a,^; y, 4x{l-a;)},
when y = a + ^ + ^, R{x)<^, |^|<1, | 4.* (1 -a;) | < 1.
26. Give the definition of V (z) by means of an infinite product and prove that it
agrees with the equation
r{z) J
where the integral is taken along a path which encloses the negative real axis and inter-
sects the positive real axis.
Prove that
2Tri fc+ix
~= T{t)z-*dt, (c>0),
^ J C—icc
where the phase of ^ is between — ^tt and -t-^Tr.
27. The generalised Riernann ^-function is defined, when ^ ( - j > 0, by the relation
, 2r(l-s) r , ,,_! e-«^ ,
t(s,a,a)) = — I -I (-^)i dx,
the contour of the integral embracing a straight axis L from the origin through the point
- to infinity, and ( — ^)1~^ having a cross cut along the axis L and being real when a; is
real and negative.
Obtain an expression for the continuation of the function for all finite values of \a\.
Shew further that, except for particular values of a/co or s, the equality
™-i 1 >/ V '^s'n(a)rd'' x^-n
^=0 (« + '*<») "=o nl \_aaf' l-sJx=m (0 ^^ ^^ terms of the values of cf) (t) at the zeros and poles of f{t) which
J c J{t)
fall within C.
It f{t) = fig {t) — 7/ where ^(0) = 1 and g{t) has no singularity within a small circuit
round t = 0, prove that/(^) has two zeros (say t = Xi, x-^ within the circuit, when \y\ is
Sufficiently small. Shew further that
(Pix,) + cf> {X2) = %4> (0) + i A^y\
n=l /
where nAf^ is the coefficient of ^2»-i in the expansion of j /\(m in powers of t.
33. Extend Cauchy's Theorem of Residues, so as to establish the following result : —
If T' be a simple contour in the plane of a variable t, and Uhe a, simple contour in the
plane of a variable u, etc., and if n analytic functions f{t,u,...z), (f){t,u,...z)..., of the
n variables t,u, ... z, have no singularities or zeros for values of t, ti, ... z, on these contours,
then
(27r^•)«,^j^^•••jz/(«,^*,...^)0(^,«,...2)^(^,«,...^)
is an integer.
34. Prove that the integral I dt is an analytic function of x in any region from
J x-\-t
which the negative real axis is excluded.
35. Shew that the integral
f{t,z)dt
will converge uniformly for a given domain of values of s, provided a function <^ {t) exists
independent of z and such that I \l except
for a simple pole at s= 1.
38. By considering the expression
„=o n\ 1 + 2™ .3;'
or otherwise, shew that if x~c be a singular point of a function f{x) represented in a
certain region by an expression Fix) it may happen, not only that F{x) and all its
diflFerential coefficients are finite at that point, but also that the series
I ^F(-){c).{x-cY
converges for all finite values of x.
39. Prove that the series
i (l+?i2 ^-2) -1(1+^2 + ^2)-!
TO = 1
converges for all real values of x, but represents a function which cannot be expanded in
positive powers of x.
Shew further that when x lies between 1 and ^^2 it can be expanded in powers of x
and x~^.
40. If the triangle formed by z^, Z2, S3 does not enclose the origin the series
00 00 00 1
2 2 2
1 1 1 imiZi+7n2Z2 + '>n3ZsY
is absolutely convergent when p>3.
41. By considering the behaviour of both sides of the identity
\_Z-Zi Z-Zz Z — ZnJ
where /(s) = {z- Zi) {z — z,^...{z — z^), on passing round an oval path on the Argand diagram
(or otherwise), prove that an oval path, enclosing all the roots of f{z) = 0, necessarily
encloses all the roots of /'(«) = 0.
42. Shew that the equations
e* = ax, eF = ax'^, (F = ax^,
where a > e, have respectively (i) one positive root, (ii) one positive and one negative root,
and (iii) one positive and two complex roots within the circle | ^ | = 1.
MISCELLANEOUS EXAMPLES 801
43. li f{z) denote a single-valued monogenic function, and Rifizj] be a rational
function of f{z), shew that a pole oi f{z) is equally a pole of i? [/(«)], unless it is an
ordinary point.
What statement can be made for an essential singularity of f{z) 1
44. - Find for what values of z the series
00 g — im
represents a monogenic function of z, and for what values its diflferential coefficient is
represented by the series of the diflferential coeflficients of its terms.
46. Rearrange the series
in powers of {z-^i), and find the value of z of greatest modulus on the circle of con-
vergence of the new series, and the sum of the series for this value of z.
46. Investigate the convergence of the series
00 ^™ + 2
^^ n=i (a»-t-l)K + 2)'
where a is a positive integer > 1 ; also the region of existence and the singularities of the
function represented by it.
47. Prove that, when n is an integer,
{z + ix cos a + iy sin a)"
n n
= Ao+ 2 ^^{(^■^-f-y)™^-(^>-«/)™} cos ma -j-^■ 2 A^ {{ix + 2/)"" -{ix-y)^} sin ma,
m=l m=l
where ^™ = — ^ ,-. =-^ ^—,
'" 2» {n + m)\ dz^^'^ '
r^ being equal to x^-{-y^-\-z^ and being treated as a constant in the differentiations.
Deduce the expressions for the 2n + 1 complete spherical harmonics which are of
order n.
48. If f{z), (j) (2), and -r-p. ^^ ^^^^ regular functions of the complex variable z in the
vicinity of the origin, shew that
for sufficiently small values of \z\.
If i z I < >/2 — 1, shew that
Ana ^^^'= ; (-)""' ^•4- (2^-2) fjz_\^n
\^l-z) nil nl S.5...{2n-l)\l-zy '
1 '^ A
49. If • — - is expressed in the form 2 , n being a
nL!;+tau2^ '^ '^-ftan2— — -
r=i\ 2n+lJ 2n+l
positive integef, shew that
Ar= -^ ^ Sm2 COS^" '^ rr -^ .
271 + 1 2n + l 2n + l
50. Shew that a straight line can be drawn in the plane of the complex variable^/
which divides the plane into two regions in one of which the function
.ziz-l) 0(0-1) (.-2)
■^"2! ~ 3! "*"••■
is everywhere zero, and in the other of which it is everywhere infinite.
P. F. 51
802 MISCELLANEOUS EXAMPLES
51. A regular function f{z) has zeros ai, ..., a,„ of orders ai, ..., a^ and it has poles
Ci, ..., Cm of orders 71, ...,7n within a given simply-connected region, which contains no
other singularities of the function ; also f {£) is regular and has no zero along S, the
boundary of the region ; and R (z) is any function of z which is regular within and along S.
Prove that
m n 1 f f (z)
^2 a,. R (a,.) - ^^y,R{c,)=^.\R (z) ''j^ dz,
the integral being taken along the contour.
Apply this result to prove (or otherwise obtain) the theorem that, if /(s, C) be a regiolar
function of its arguments within finite domains round the respective origins -and ii f{z, 0)
has the origin for a zero of order n, then the equation
/(2, = 0,
regarded as an equation in 2, has n roots which vanish with {" and which are the roots of
an equation
2» + C]2'^~' + --- + t« = 0,
where ^1, ..., ^„ are regular functions of f all of which vanish when ^=0.
52. By consideration of the expression 2 —^ . ^ , or in any other way, prove that
n=\ "^^ I — Z
the function defined by the series 2 a^z^, where a„ denotes the sum of the reciprocals of
n=l
the squares of all the divisors of n, exists only within the unit circle whose centre is the
origin.
53. Let (f) (x) be an analytic function defined by a series of powers of x, of unit radius
of convergence. The points x = a and .v=b are singularities of
and ^ - ^ cos ?2-7rA',
n'' (smh 2?i7rp - zmrfj,)
where ?i = l, 3, 5, ..., is approximately equal to
^ {l-ex^+4x^)-^il-2x);
128fi^^ ' 320
and obtain an expression for the difference when - is large, in terms of the roots of the
equation
sinh 2=2.
51—2
804 MISCELLANEOUS EXAMPLES
III.
62. Shew that, when m and n become infinite together in a finite ratio,
s
™ / 2 \ _ /»*V sin z
63. Discuss the convergence of the infinite product
w=i \ c„y
where c^, C2, C3, ... have the single limiting point 2=co . ■
Form an integral function whose zeros are the points log 2, log 3, log 4, ....
64. Assuming 2 | a„ | is convergent, deduce the form of the product
Prove that, if | 5- 1 < 1,
n (1 -y2«) n {(1 + j2«-i^) (1 + ^2n-i/^)} = i +2 (x-^ +— j q^.
65. If ? and m become infinite in such a way that — = jo, where jo is a finite quantity,
shew that
2(«2+t;2) n' i 22 [ = (cosh2y-cos2M)p'^ ,
where the accent in n' denotes that the term corresponding to 7i=0 is to be omitted.
66. Discuss the problem of finding, where possible, the most simple type of integral
function whose zeros are given by
n
r=l
where m^ can take all positive integral values (zero included) and the a's are general
complex quantities.
67. If 0(.)=^_n{(l-?).^}.
and if the values of (f) (x) and its differential coefficient for x=^ he denoted by a and b,
prove that
bx
J. {0-■;^i)'"i=^"° *<-+*>■
68. Discuss the problem of constructing an integral function whose zeros are known.
What is meant by the class of a function ? Is it possible to determine it without knowing
its zeros ?
Prove that, if F (2) be an integral function of class p, and have all its zeros real, then
F'{z) has at most p imaginary zeros, if p be even, and {p - 1), if p be odd.
MISCELLANEOUS EXAMPLES 805
69. Shew that the function
r(g)r(a)
T{z+a) '
which, when a is positive, can be expressed in the form
°° R
where ^,(-l)-(.-l)^.-2)...(a-.)^ ^_^_
can, when a is negative, be expressed by the series
,,=0 V + ^i "^^j'
where (r„(2) is a polynomial of degree {v-l)\az, defined by the equation
whUe 0{z) = (\^~\U + ^-^...(\ +
aj\ a+lj V a + v-ly
and where v is the integer next greater than — a.
70. If ai, a2, as . . . , he a.n assigned simple sequence of zeros whose moduli ultimately
increase without limit; if the real part of a„, for n=l, 2, 3..,, is positive; and if
2 1/1 a„ I** is convergent, shew that the product
is convergent if o-^2p > 2, where
71. Construct a function f(x) which has a pole at each of the points «=1, x=2, ...
and no others, and such that/(.'«7) — .r cot 7r^-*-0, at each of these points.
72. Discuss the continuity of the function
and prove that, whatever be the character oif{z) and g{z), the function
^ {zg (?) -f{z)} + {/(.) - g {z)} 4> (z)
represents the function f{z) within the circle | 2 | =1 and the function g (z) without that
circle.
73. A finite region of the plane being given which does not, include the part of the
real axis between z = l and z=+cc, obtain a series of polynomials which converges uni-
formly over this region and represents (1 — 2^)"* therein.
74. For the region between the two circles
(x -a){x-b)+f= 0, (x - a') (x - h') +y'^ = 0,
where a, b, a', b' are real and positive, and a I ^ (2) I
is satisfied.
Shew that f{x) can be expanded in powers of 6 {x) in the form
00 00 7?
/(^)= 2 Ar,e-{x)+ 2 -^ ,
w=o 51=1 ^ K^J
where ^ - ' ' f^'^''^'^''
Bn = ^^f{.^)e^'-'{z)6'{z)dz;
and shew how the coefl&cients A^ and 5„ can be evaluated when the function f{z) has no
singularities in the interior of the curve s except poles.
76. Obtain the theorem known as Mittag-LefHer's, for the expression of a single
valued monogenic function whose singularities have only the infinite point as their point
of condensation, as a series of functions each with only one finite singularity.
Prove the ordinary formula for cot irz as an infinite series of rational functions each
with one pole, pointing out the properties of the trigonometrical functions which you
assume. Prove the corresponding formula for the logarithmic differential coefficient of
the Gamma function r(l + 2).
77. Explain what is meant by a branch and a branch -point of an analytic many
valued function. Illustrate your remarks by consideration of the following functions :
1or(1— ^) /riog(l — ^))
log(l-^), J{\og{\-x)}, -^V~' Vl \ }•
In each case enumerate all the branches of the function and the singularities attaching
to each branch, and shew how to pass from any branch to any other branch by the
description of a suitably chosen contour.
78. Defining the principal value of log 2 as that value whose amplitude lies between
- TT and TT, prove that the coefficient of i in the expression
log(l+^■^)-log(l-^■^),
where r<\ and each logarithm has its principal value, is that value of
/2?-cos(9'^
arc tan — ^ -,
\ 1-r
which lies between — -jtt and -f^rr.
MISCELLANEOUS EXAMPLES 807
IV.
79. By the relation
with the customary significance for Z and z, shew that a family of circles in the Z-plane,
having the origin for a common centre, and a family of straight lines, concurrent in the
origin, are transformed into a double family of confocal conies in the s-plaue. Draw the
family of curves which are the representation, in the .Z'-plane, of the circles
I 2 — 1 1 = constant.
80. The interior of the circle j s j = 1 is to be conformally represented on part of the
w-plane by the relation
log(l-ag)
''-log(l-.)'
where a is a real constant lying between and 1 ; indicate the part of the ■z<;-plane
required for the purpose.
81. Find all systems of values of m, y, for which sn2(w+w) is real, where u and v are
real and 0a.
Taking x=- (a + -
2 \ a
so that vP'=-z^-'±xz-k-\.,
dz
1 (zH
2117 J u
shew that „ . ,
2^7^ J u
is a solution of Legendre's Equation
fJ2 p dp
(l-^2)^.-2^^' + »(v^ + l)P„ = 0,
the integral being taken round a closed contour including the points z = a, z=- , but
excluding the origin.
Assume (after verification) that
dz \^ u^ ^ " II ) { u^ u^ It]
87. Writing .^=Jr+zF, where X and Fare real, and taking
Z= sin z,
determine a simply-connected region of the plane of z which is transformed conformally
into the half-plane Y> 0.
88. Prove that the relation
2 = aCOs(|3log.Z'),
a^ + lfi
where a^ = a^ — h'^, cosh/37r = -5 — j^,,
a^ — 0^
gives the conformal representation of the interior of the region' boimded by the ellipse
-2 + ^2 = 1? aiid two lines joining the foci to the extremities of the major axis, upon the
interior of an annular region in the .^-plane bounded by two concentric semi-circles and
two segments of a diameter.
89. Find the area on the z-plane of which the upper half of the w-plane is the
{z — c\^
conformal representation, when w and z are connected by the relation w^l \ .
If ?;•= - ic cot ^z, shew that the infinite rectangle bounded by .^"=0, a'= tt, ^=0, y = oo on
the 5-plane is conformally represented on a quarter of the i'j-plane.
90. Shew that the most general representation of the interior of a circle of unit radius
upon itself is expressed by the formula
z' = ize^'' + ixe'^)/{zixe-'^ + e-'''),
where fi is real and positive but less than unity. Obtain the most general conformal
representation of the interior of a square upon itself
MISCELLANEOUS EXAMPLES 809
91. Prove that the surface of revolution engendered by the revolution of the tractrix
about its base can be conformally represented upon the plane of 2, in virtue of the fact
that the square of its arc-element can be conformally represented in the form
where x ranges from to 27r, and y ranges from 1 to + 00 .
Discuss its representation upon the s'-plane, when z and z' are connected by the relation
/ = sin2{i(s-^•)},
indicating what part of the s'-plane is covered by the conformal representation.
92. Develope the Schwarz-ChristofFel formula for conformal representation in detail
in the case when the polygon is a triangle whose angles are — , — , — , discussing the
Ji S Xi
character of each variable as a function of the other.
93. Obtain the conformal representation of the interior of the equilateral triangle
whose vertices are 2 = 1, z — i J^, z= — I, upon the upper half of the plane of ^, expressing
each of the variables z and f explicitly in terms of the other.
94. State briefly in precise analytical terms what you understand by a closed oval
curve everywhere convex and with a definite tangent. For such a curve drawn in the
plane of the complex variable z, an analytic function exists, single-valued and finite,
and having its imaginary part positive within the oval, which is real and has one pole of
the first order on its perimeter. Explain how this is to be proved.
Find the function for the conformal representation of the part of an infinite plane
which is exterior to two intersecting circles upon the upper half of another plane, verifying
that it has the properties desired.
V.
/T'
95. Assuming that -=. is real and positive, establish the formula
9^ ^^ ~^ " ^ 1 _„2»-l '
determining the values of x for which it holds ; and find the corresponding formula for
any finite value of x.
96. Prove that
2Kx . -r/l + o2™-i\2 / l-2o2»cos2^-f-o*» \"1
^^^r = '^^^TLVTT?^j U-2g^»-cos2^^-fj^»-2.jJ'
a :Anx)n\( ^-^"^~y (l-2g'>sin.^ + y^-)^ I
(i-sm^;ii|^^ j_l_^2» j (i_2^2«-ico8 2^-i-?*«-2)J'
1 /I \^
97. Prove that, if ^' = - ( - - a ) , «. being positive and less than unity,
and that
2Kx
1 — sn
snH^= ^'"^
■(l-f-a2)(l+2a-a2)'
and prove that the value of sn^ |^ is obtained by writing - - for a in this expression.
810 MISCELLANEOUS EXAMPLES
98. Discuss the functional relations connecting the pairs of the three quantities
(i) the cross ratio of the roots of the quartic polynomial f{z\ (ii) the absolute invariant J
of this quartic, (iii) the ratio r of two fundamental periods of the elliptic integral
99. Obtain a formula of reduction for Jsn» udu ; and thence shew how to perform the
integration when n is an even positive integer.
Establish the formula
sn^ u du
(i+F)t-/;,-^
=1.
[-\-CQ.u)diV?U
100. Prove that
?5_^=-y&'-l{f(^i-^)-C(tt-^-2lX')-f(2^X')},
en w
the f -functions being formed with the periods 2^, AiK'.
101. Establish the identity of the infinite series
1 +2 2 ( — )" q"^ cos 'inx
1
with the infinite product
(?n(l-2j2«-icos2A' + g^™-2),
1
where G = li.{\-q^'^).
1
Either of these expressions being denoted (as in Jacobi's lectures) by ^ (^), prove that, if
Ig-I^ <|e*^| < l^pi,
2Z#5(0) , „^ „
• — t^ , = tto + 2 2 (X„ cos 2%^,
ir ^ {x) 1
where a« = 2 i ( - 1)™ j(»^+i)(2»+m+i).
102. If Mi + M2 + W3 + ^4=2^, shew that the anharmonie ratios of the four quantities
sn 2fi, ..., sn Ui are equal to those of the quantities sn (? 0, and | /(m) j if | g|^< | g/^) | <1,
1
h
<
?
117. Prove that, if | 5- 1 < 1, the series
^1 (^) = 2 2 ( - 1)" qin^W sin (2% + 1) .*',
and the product
sin A- n ( 1 - 22'2» cos 2.^ + ^*"),
differ from one another only by a constant factor.
Prove also that, \i q = e K ^
m being any integer.
118. Defining the Jacobian theta-functions by the equations
- » -00
i(9i(A')= 2 (-l)'»^(m+i)2e(2m + l)a;i^ ^2 (^') = 2 ^(»i +*)''' e(2m + l)a;t^
00 — QO
prove that 6^^ 62, O3 are even functions of .r, while di is an odd function of x ; that ^q and
^3 are periodic with tt as period, while B^ and ^2 are periodic with Stt as period ; and that
all four functions are pseudo-periodic with i log g as a period, two of them with - e^^t as
1 . '
a factor, and the other two with — e^m ^s a factor.
Prove that
^3-^ (0) 6s {X + i/) Os {X -I/) == 6i (x) ds' (y) + e;^ {x) e,^ {y)
=^e,^x)e,\y) + 6i{x)e.i{y);
and deduce from these equations similar exjiressions for
ei{Q)6,{x+y)e,{x-y)
for r = 0, 1, 2.
119. With the notation
^{x)= 2 (_l)»j»2g2ma;i^
^2 {x) = 2 ^i (2-1 - 1? e(2« - 1) a;i,
establish the periodic and pseudo-periodic properties of the ^-functions.
Prove also that the expression
4 4
n ^2 (■«?«) + n .^3 {x^)
1 1
is unaltered when jti, X2, Xs, x^ are replaced by
^{Xi + X2 + X3 + Xi), |(.^i-t-.^2-*'3-'*4)5 | G^'l - .^2 + ^3 " -^'4), i i^l' •'^2 — ^3 + ^i) ',
respectively.
814 MISCELLANEOUS EXAMPLES
VI.
120. Examine under what circumstances p {ku) is a rational function of p (u), k being
a suitable constant multiplier ; and if ?7,i be a function of u defined by the equations
^1 = 1, U^=-p'{u\ U^^, = ^{p{u)-p{'>iu)},
prove that f2m + i '^^^ C^2m/P' ^^^ rational integral functions of p (u), ^g^ and g^, with
integral coefficients, and that
If IT — IJ TT 7^7 2 _ TJ TT TJ 2
^ n — in'^ n-\-in ^ n+X'^ n—\^ m '^m + l ^ m—l "-^ » >
where m and n are positive integers.
Prove that, if p (w), g^-, and ^3 are rational numbers,
p{nu)ei> 63), prove that
{^(M+i-)- 63}!^ («-.') -63}'
{ ^(«)-g2} {^W-e2}-eie3 - 262'
LIP W- 63} Wky)~e^]-ex^i
-2622-12
-2.32 J
<
iu
{PW
-p(a)}2
:-['
dx
127. Prove that a doubly-periodic meromorphic function of %i can be expressed
linearly in terms of functions f(?« — a), f'('w — a), ..., where f (?<) = — -j-^-
Evaluate /
J
128. Shew that the equation
■ -r — ^
J xo Jf{x)
where /(a") is any quartic function of x, and Xq is a root oi f{x), is equivalent to the
equation
_ /' (.To)
^-^■« + 4{^(.)-Jj/"K)}'
where the doubly -periodic function ^ {z) is formed with the invariants g^ and g^ of f{x).
If Xq be not a root of /(^), shew that the last equation must be replaced by
^^^ , /^ (^0) F (^) + i/ (-^0) W {^)—hf' (^0)}+ 2V/(^0)/" (^0)
2 {^(2)_ Jj./"(^o)F--^L/(^o)/"(^o)
129. Shew that, if n is an odd prime, the value of the elliptic function of the wth
part of a primitive period, e.g. ^ ( — J , may be obtained by solving an equation of degree
71-1-1, and ?i-|-l equations of degree n — \.
816 MISCELLANEOUS EXAMPLES
1 30. Starting from the definition of o- (z) as an infinite product, prove the formula "
a {b + c) cr{b — c) a{a + d) a- {a — d) + (r{c+a) a- {c — a) a {h -\- d) a- {h - d)
+ (T {a-\-h)
,.,, , ,. {g>(^)^(.y)-ig^2} {g>(^-)+^(y)}-i^'(^)F(y)-^^3
Prove also that
0'3
3C(3«)-9a«)=^._.^^p_^^^_^^^^2;
and shew that n^ (nu) - n^ ^ (it) can be expressed in terms of p and ^' only, n being an
integer.
138. Prove that
2p' (2«) f (u) = {^ (?^) - p {u + o))} {p (u) -p{u + 0)')} {^ (u) -Piu + o)")}.
r=4 1
139. Express (62 - 63) n {p (t<,.) — ^il +two similar terms, by means of Jacobi's elliptic
r=l
4
functions; and hence (or otherwise) shew that, when 2 ^^,. = 0, the expression is equal to
r=l
- (61-62) (^2- 63) (63-61 )•
140. Shew that
2o
.•2i7(m+(o)/)h
'(4Kt)]
where 2a> is a period of p {u) and m is an integer, is the mth root of a rational integral
function of p (u) and p' (u).
141. Obtain a general formula for p (nu) in terms of p (ii).
The periods of p (u) being 2co, 2a)', and a, b, c, d, denoting the quantities
prove that a^ + eb^ + e^c^ = 0, e'^b^ - ec^ + d^ =0,
where e is an imaginary cube root of unity.
142. Specify briefly the descriptive properties characterising the elliptic function
P (•?«), and prove that the function is determined thereby.
Shew that the function (p (u) = {p (u) — e,}"^ is a single-valued function; obtain its
periods, and the expression of cf) {u+v) in terms of (f) (u), 0' (m), (v), 0' (v).
Find the ^-function having the periods of (r{u- 6)],
and the algebraic relation connecting these derivatives. ■
145. Prove that every elliptic function can be expressed linearly in terms of ^(m - aj),
^(w — 02), ... and their differential coefficients, where C(^) = o"' {u)/(r{u).
Prove that
1 yw+FW _ &'Xv)+&'H ]
146. Establish the formula :
= -C{'^-u) + C{w-v)-i-C{v)-C{u).
1, ^W, F(«)
I, Piw), p'{w)
147. Express
= -2a- {u-v)cr{v -to)cr (w-u) cr {u + v + w)j a^ (u) a^ (v) a^ (vj).
1, p(a;), p(^), p'{a;)
1, ^(3/), PH^), P'{y)
1, ^(^), P(2), F(2)
1, p{u\ P(«), p'{u)
as a fraction whose numerator and denominator are products of a--functions. Deduce
that, if a = p {x\ /3 = p (3/), 7 = ^ {z), 8 = p {u), where x+y + z + u = 0,
(62 - 63) {(a - ei) O - ei) (7 - ei) (S - ei )}* + (63 - ^i) {(a - eg) (/3 - 63) (y - e^) (8 - e,)}^
+ (ei - 62) {(« - 63) (3 - 63) (7 - 63) (S - 63)}* = (62 - %) (63 - 61) (^1 - 62).
. 148. Denoting the roots of 4^^ — ^2^-5'3 = by ei, 62, 63, prove that
where I, m, n=l, 2, 3, and the summation extends over the three corresponding terms.
149. Prove that, for any three arguments u-^, u^, u^,
C(%) + CC«2) + t(%)-CK + W2 + %)
_2 W (^1) - P (^2)} W K) -
)-^(ia, + a,') = 2 {(61-62) (61-63)}*,
^' (i») = - 2 {(61 - 62) (61 - 63)}^ {(61 - 62)^ + (61 - 63)^}.
153. Shew that
r(.)=6
MISCELLANEOUS EXAMPLES 819
a- (z + a) a- (z — a) a- (z-hc) a (z — c)
0-* (z) 0-2 (a) 0-2 (c)
where P {a) = {^92)^-, ^ («)= -(tV5'2)*-
154. State the properties of the eUiptic function p {u) which prove that there is a
single-vakied function a (u), such that a^{ii) = p (^) — ^i, and ua{u) = l when u = 0.
Defining similarly b (w) = (p (u) - e^^ , c {u) = (p (w) — gg)^ , prove that
a (u) b {v) c (v) — a{v)b (u) c (u)
Shew also that
a{tL + v)=-
a^ {v) — a^ (u)
a{u + a>)a (?<) = a' {a) = — or'^ ( i^ ^j,),
2a (m) 6 (m) c{u) a (2u) = a^ (u) — (X* (^ co),
'<^ fl 1
-^ a{u)\- dzo = log [^ M {6 («) + c («)}],
da{u)
where a iu) = — j^ .
du
155. From the theory of doubly-periodic functions, or otherwise, obtain the formula
(.)
Shew how to express the points of a plane quartic curve, which has two double points,
by means of elliptic functions.
^(»)-^<»-)=-.s{frgl^
VII.
156. Obtain Euler's relation C+ F=E+2 connecting the numbers of corners, faces
and edges of an oi'dinary convex solid bounded by plane faces ; and extend this result to
the case of a solid for which the surface is not simply connected.
A closed rectangular box has a partition lying midway between two opposite faces, and
this partition is pierced with two holes. Discuss the connectivity of the inner surface.
157. Prove that on any closed sui-face in space (or on a Riemann's surface) a system
of closed curves can be drawn, so that (i) it shall be possible to pass from any one point of
the surface to any other by a continuous path which does not cut the closed curves, and
(ii) any two paths so drawn between two given points shall be continuously deformable
the one into the other without crossing the closed curves.
Construct such a system of closed curves for a three-sheeted Riemann's surface with
two branch lines at each of which the three sheets are connected cyclically.
158. A table consists of a rectangular parallelepiped resting on two other rectangular
parallelepipeds, placed vertically, and each pierced with a hole. Upon the table is laid a
book, and upon this a much smaller book. Estimate the connectivity of the surface of the
whole I'esulting solid, each book being regarded as a rectangular parallelepiped.
159. There are n rings placed in order, each connected to those on either side by a
cylinder, whilst the two rings at the end have each a point boundary. Find the con-
nectivity of the surface so formed.
There are n rings placed in order, each connected to those on either side by a cylinder,
whilst the first and last rings are connected to each other by a cylinder. To the surface
so formed, a point boundary is supplied. Find the connectivity of the surface.
52—2
820 MISCELLANEOUS EXAMPLES
160. Prove that, if an anchor ring be hollowed out and the exterior and interior
surfaces connected by a hollow cylinder, and if a point boundary be supplied to the surface
so formed, then its connectivity is 5.
161. A surface of connectivity n has h boundary lines. Shew that it is impossible,
without dividing the surface, to make more than ^{n — h) cross-cuts, each of which passes
from a point of an original boundary to another point of the same boundary.
162. A variable u is defined by means of the relation
/"« dx
where y^ = Ax^- g^x- g^,
and the inversion of this relation is expressed in a form
apply Abel's Theorem to shew how an expression can be obtained for Q{u + v) in terms of
Q (w), Q' {u), Q (v), Q' (v). Find also the periods of the function Q (u).
Construct the integrals that remain finite on the Riemann's surface associated with
i/i = 4x^- g2X- gs.
163. The equation f{w, z) = is algebraic in w and z ; and it is satisfied hj w = a,z=a.
Shew that, when z=a + z', where \z' \ is small, then values of w are given by a + v/, where
I ti^ I is small, and where w', if not a uniform function of /, belongs to a set of values the
members of which interchange cyclically when z describes a small circle round a.
Obtain explicitly the branches of the function w, as defined by the equation
w^ + 3wz + z^ = l
for values of z, (i) near the origin z = 0, (ii) near the point z=-l.
J (■
164. Shew that the integral
{{x - ai) {x - a^) {x - as) {x - ai)} ^ dx
is transformed to the integral
by the relations
(a2-ai)(^-a4)' («3-«i)(«2-a4) '
and obtain an expression for the general value of the former integral.
165. Prove that every integral of the first class associated with the equation
iv^ + z'^ = l
is of the form
\aw + Bz^C)'^^^,
where A, B, C are arbitrary constants ; and construct integrals of the second and the third
classes, associated with the same equation.
/<
166. Obtain the integrals of the first kind connected with the equation
vfi — z{z—l){z~a) (z — b),
where | a | > | 6 | > 1 ; and shew how to obtain the addition -theorem for the functions that
arise in the inversion of the integrals.
MISCELLANEOUS EXAMPLES 821
167. For the equation
y^ — byx^ + 4^ = 0,
find the form of the everywhere finite integrals.
168. Determine the genus of the equation
y'^ = x {\ — xy.
Find a system of integrals of the first kind, and also an elementary integral of the
third kind with its infinities at the values x = 0, x = l.
169. Construct and dissect the Riemann's surfaces associated with the equations
(i) w^=z^{z'^-\\
(ii) v^ = {z-aif{z-a2f{z-a^f,
(iii) -?«;*= (0-ai)2(0-a2)3 (2 -a3)3.
Give two dissections for the last surface, one of which does not involve any cut in one
of the sheets.
170. Construct a Riemann's surface on which the function w defined by the equation
2<^-(22 + 3)w2 + l=0
can be exhibited as a uniform function.
171. Explain in general terms the principles of the theory of the dissection of a
Riemann's surface, to render the surface simply connected, and the part which the
number of everywhere finite integrals upon the surface plays in the number of necessary
dissections.
Shew how to find the number of everywhere finite integrals associated with an equation
y»» = (^-aj)»i(^-a2)"2 ....
Find these integrals in particular for the equation
{x — a){x — h)
r=
and dissect the surface.
{x-c)(x-d)'
172. Describe the character and position of the infinities of the integral
'y^ + ax'^ + Ax + By + C
/^
{2y^-x^-l)y
where w is an imaginary cube root of unity, and x, y are connected by
x^+x'^y'^+y'^ = x^-\-y^ + ^;
and find the sum of its values extended from the points where Px-\-Qy + R=Q to the
points where P'x+Q'y + R' = 0.
173. Construct a Riemann's surface to represent the equation
{x-l)y^ = a^{x + lf;
draw cuts reducing it to a simply connected surface ; and construct an Abelian integral
of the first kind associated with the equation.
174. Discuss the general pharaoter of the Riemann's surface which represents the
equation
{w — z) (w^ — z'^) (w^ — 2^) = 1 ;
and prove that its genus is equal to 4.
822 MISCELLANEOUS EXAMPLES
175. lu the whole of a Riemann's surface of n sheets, the total number of branch
points is s ; and the numbers of the branches of the represented algebraic function,
which interchange at the branch-points, are m^^ m^, ..., mg respectively. Prove that the
connectivity of the surface is
( 2 9n,.^-(2w + s-3),
and that its genus is
Prove that the genus of the Riemann's surface associated with the equation
w^^=A{z-af{z-hf{z-c)\^
where a, b, c are unequal constants, is 7 ; and indicate the relations between the sheets
of the surface at each of the branch-points.
1 76. Indicate various classes of functions of position on a Riemann's surface of genus
p, explaining specially the characteristic properties of the functions which usually are
called of the first kind, the second kind, and the third kind, respectively, as well as of
adjoint polynomials j and prove that an adjoint polynomial possesses 2p — 2 zeros on the
surface.
At a set of m among these 2p—2 places, q other adjoint polynomials vanish together :
at the remaining 2p — 2 — m places, q' other adjoint polynomials similarly vanish together.
Prove that, on a general Riemann's surface,
q' — q = m—p + \.
177. Find the genus p of the relation
Construct a rational function associated therewith with p+\ arbitrary poles, and
obtain the forms of p everywhere finite integrals.
178. For the Riemann's surface associated with the equation
^y -f Ms + ^4 = 0,
where %, % are homogeneous polynomials in x, y respectively of orders 3 and 4 with
general coefficients, construct a set of linearly independent integrals of the first kind
and an elementary integral of the third kind whose infinities are at ^=0, ?/ = 0.
Find, save for an additive constant, the sum of the values of the integral
/
^-dx
X
at the intersections with the given curve of the line Ax-\-By-^C=-'d^ expressed in terms
of -4, 5, C, and the coefficients of the curve.
179. Construct a Riemann's surface suitable for the representation of a function y
given by the equation
Make a series of cuts which will render the surface simply connected.
180. Shew that the genus of the equation
y'^-\-y{,x,y)''-\{x,y)^ = .
is unity ; and construct an integral of the first kind associated with the equation.
Similarly for the equation
/=(l-^2)(l_^2^2)_
MISCELLANEOUS EXAMPLES 823
181. Obtain the position and character of the branch places of the Riemann's surface
representing the equation
and the forms of the everywhere finite integrals.
182. For the fundamental equation
y* = .K^ + ^ + 1,
describe the behaviour of the integral
/-[|--5^.]^
and find the sum of its four values integrated from the four points where P^^x + §oy + -^0 = 0,
to the four points where Px + Qi/ + R = 0.
183. Prove that the equation of a curve of order n, having ^n{7i — 3)-l double points,
can be transformed birationally into the hyperelliptic form
where/ is a polynomial of degree 5 or 6. Transform in this way the equation
f = x^'{x^--3x+2).
184. Prove that the equation
fix, y) dx
wherein y^=^- g%oc-g^^ defines x^ y as single-valued meromorphic functions of %.
If y'^=a?'X^-\-4tbx^-\-Qcx"-\-'^dx-\-e, and R{x,y) denote a rational function of x and y,
find three integrals
R {x, y) dx
/^
respectively (i) everywhere finite, (ii) algebraically infinite to the first order bvit not
logarithmically infinite, at .■v = |, y = r), (iii) logarithmically but not algebraically infinite
s,t x—^, y=rj, andat.r = co, y = ax'^; and shew that every integral jR(x,y)dx is expres-
sible, save for rational functions of x and y, in terms of integrals of these three forms.
185. If JO and n be positive integers, yP = l + aj^x + a2X^ + ...+anX"; the right-hand
side having no repeated roots, prove that
v= I dx/yP ^
J
is an elliptic integral, provided that p and n have the greatest values consistent with
the inequalities
-+ ->:! and -+ ->1. .
p n n p
Determine the possible values of p and b.
Shew that p (v) can be expressed in the form
where t>=0 corresponds to ,^;=0, y = l.
Completely determine the invariants ^2 ^^'^ ffs "^ *^® cases ji9 = 2, n = 4, and p — 3, n—3.
824 MISCELLANEOUS EXAMPLES
VIII.
186. If fix) be a quintic polynomial in x, and
investigate the character and general form of Xi + x^a.^, fi, function of Uy and U2 .
187. If 3/^ be a quintic polynomial in .r,. vanishing when x=ai and when .2; =0^2 5 and
discuss the character of XyX^ and of {(«! — ^1) («i — *'2)}* as functions of % and 'M2'
r^i 0?.^; /'■^s 0?^ /"^i xdx f'
J a^ y J a^ y J a^ y Jo
X ~~~ Xq
■(^0 Ex + F , f^^'^'> Ex+F
dx +
y
188. If A, B, C, D, E, F be constants, f{x) a quintic polynomial, (^o^o) a variable
pair satisfying y'^=f{x), and {xi), (x^), ..., be the zeros of the rational function in {x, y)
A {x-\-x^)^-Bxx^-\-Cy;^^^D,
d Y f^^^^ Ex + F , f
evaluate -^— | ax+ I
"■^^o L ./ y J
189. Quantities u a,ud v are defined by the relations
2u=r^:^dx+ri^^d,,
where X={X'~ ao) (x — aj) {x - a^ (x — a^) (x — a^),
T is the same function of j/ as X is of x, and the constants ao, «!, a2, «3, a^ are real,
unequal, and (so arranged) are in descending order of magnitude. Prove that any
rational symmetric function of x and y is an even quadruply-periodic function of u and v,
and that, in particular,
{Ur - x) (a,. - y)
(for '/■=0, 1, 2, 3, 4) is the perfect square of a quadruply-periodic function of u and v, which
is even for even values of r and is odd for odd values.
Writing p^? = {a^ - x) (a,. — y),
(«™ - «.) Pmn -Pm {j^i + J^ ) ' Pn[ ^^ + ^^ ) ,
obtain the types of quadratic relations connecting the fifteen functions ; and prove that
Pi , Pm ^ Pn I )
Plrj pmri Pnr
Pis 5 Pms ) Pns 1
where I, m, n, r, s are any arrangement of 0, 1, 2, 3, 4, is constant.
MISCELLANEOUS EXAMPLES 825
/:
190. If the curve F {z, m) = be of degree m, shew that every integral of the first
kind is of the form
Fu (^, m) '
where Q (s, u) is a polynomial in z and ii of degree (m- 3) at most.
If the curve F {z, u) = Q have no multiple points except such as have distinct tangents,
shew that any multiple point of order q on the curve i'' is a multiple point of order q—1
on the curve Q.
Find the integrals of the first kind for the curve
z'^-u'^ + ahu^O ;
and discuss the transformation of this curve to the hyperelliptic form.
191. Shew that every uniform function of position on a Riemann's surface, connected
with an algebraic equation f{w, z) = of degree m in io, the function having infinities
only of finite order, can be expressed in a form
dw
where C^ is a polynomial in w of degree ^m — 2 having rational functions of z for its
coefficients, and ff (2) is a rational function of z.
Prove that there are integrals of rational functions of w and z, which do not acquire
an infinite value upon the surface.
Construct these integrals, when the algebraic equation is
w3 = s (l-z) (1 -az) (1 - bz) (l-cz),
where the constants a, b, c are unequal to one another and no one of them is either zero
or unity.
192. If an analytical correspondence be set up between two variable points x and y
of a non-singular Riemann's smface, of such a nature that to every point x there cor-
respond m variable points 3/1,3/2) ■•■■>ymi distinct in general from x, prove that
m
2 %(,yj)-fy%(.^) = Ji, (-^=1, 2, ...,/•),
i=l
where y is a certain positive or negative integer, Jj is a quantity independent of x, and
%i, U2, ..., Wp are p independent and everywhere finite integrals on the surface.
Prove further that, if Cj, C2, ..., Cp be suitably chosen constants, and if
denote the ^-function belonging to the surface, then
6{ ui{y)-Ui{x)-c,] «=™ e{Ui{y)-Ui{y^)-G,)
6 K {.y) - Wi i^) - Ci} bh if) - Mi {^) - Ci} n=i B {Mi ( /) - y'i iyn) - Ci) e k {y) - Ui (y„o) - c,]
is an algebraic function of x and y ; x^, j/", denoting fixed points and x, y a. variable point
of the surface, z/i", 3/2**, ..., ym^ the points corresponding to x^.
Deduce that such a correspondence can always be represented by an algebraic function
properly interpreted.
826 MISCELLANEOUS EXAMPLES
193. If Wi, Wo, ..., u>p be the p normal integrals of the first kind on a Riemann's
surface, and 5;,(*) be the modulus of periodicity of the normal integral of the second
species to tc,^{z) with respect to the cut b^, shew that
Hence or otherwise shew that the necessary and sufficient conditions, that q distinct
points Ci, C2, ..., Cq{q^p) should be such that at a certain number of them a regular
function assumes the same value, are
=
w/(Ci), ...,Wi'{Cq)
W2'{Ci), ..., W2'(Cg)
Wp'{Cl), ...,'Wp'{Cg)
Shew also that if, for every integral of the first kind W (z),
W (ci) dci + W (cg) dc2+...+ W (Cg) dcg = 0,
then Ci, Cg, ...', Cg are points at which a regular function assumes the same value.
IX.
194. Give an outline of a proof that a potential function ti exists, subject to the
conditions,
(i) at all points within the area of a circle of radius unity, the quantities
M, ^ , ^ , — , ^-"2 are regular functions of x and y such that ^^ + ^-2 = ;
(ii) the quantity u acquires assigned values along the circumference, which are
regular functions of the position on the circumference ; and obtain the function in
the form
2 r 27r \ _^2 _ y2
""^^nj ^^^'^\x- COS ^/.)2 + Cy - sin y\rf '^'^'
where / (■v//') represents the values along the circumference. Prove also that / (yj^) can have
a finite discontinuity at a limited number of points.
Apply the transformation x+iy — -TjT. ^^ — ^ to the above integral so as to prove
that a potential function u, which exists over the whole plane and is such that its value
is unity between — 1 and + 1 on the real axis and elsewhere is zero on the real axis, is given
by - ^ where 6 is the angle subtended at the point in the plane by the part of the real axis
lying between — 1 and + 1.
195. Deduce, from Cauchy's integral formula, Poisson's expression for the value of a
developable potential function at any point interior to a given circle, in terms of the
values of the function on the circumference.
Shew ab initio in the case when u (t) is finite and continuoiis for all real values of t,
the values ^( + 00) being the same, that the integral
1 /■+" u{t)ydt
represents a developable potential function of {x, ?/), for y>0, reducing when (x, y)
approaches to {xq, 0) to the value t(,{xQ).
Evaluate the integral when u (t) = -x — -.
MISCELLANEOUS EXAMPLES 827
196. Taking as an area the whole of the plane of z with the exception of the finite
straight line joining z= — \ to ^ = 1, find a function of a, which is single-valued in the area,
is real on the boundary, and is discontinuous within or upon the boundary of the area
only like {z - i)~^.
197. Find a function which shall be regular within the circle 1^1 = 1 and shall have on
the circumference the value
a^ - 2a cos d + cos 26 + i {2a sin 6 - sin 26)
{a^-2acos6 + lf '
where | « | > 1,
198. Prove that the most general form for a function which (i) is to be single-valued
and analytic in a rectanglis in the plane of the complex variable u whose corners are u = 0,
m = 2q), u = iH, u=2a> + iH, where co, iTare real; and (ii) is to be further such as to assume
equal values at opposite points, ?i = t^, ^t = ^■A-f2«, of one pair of sides; is a series of
integral powers of exp ( — m j .
If a < 6 be real and positive, and the function p {u) be consti-ucted with the real
period 2a) and the period 2co' given by
, im , b
&) = — log - ,
IT ^ a'
find the region in the plane of z given by the formula
when ( varies in the annulus lying between two circles with centre at ^=0 of respective
radii a and h.
Shew that there is, in this annulus, only one value of ( corresponding to any point in
the region obtained.
199. Shew how to define an integral function of the two variables u, v, which shall
satisfy the equations
0(W+1, v) = (f,(u, v) = (t){u, v+l),
(}) (2i + p,v + a-) = )u/'' (f) (u, v), (p (m + p',v + a-') = /x'e^'" ^ {u, v),
obtaining any necessary conditions for the constants p, cr, p', a', X, X', p., p.'.
200. If the substitutions of an infinite discontinuous group be
t^J^ (^=1.2, ...CO),
shew that the series
2 (7^3 + Si)-''"
i=l
is absolutely convergent when m is a greater integer than unity, except for special points z.
Construct a group of substitutions of genus p, for which the fundamental polygon is
the space outside 2p circles, and the fundamental substitutions make these circles corre-
spond in pairs ; and shew that, subject to certain inequalities, the series
^{yiZ+K)-^
i
is absolutely convergent for groups of this character.
828 MISCELLANEOUS EXAMPLES
201. Give an example of a rational function R (x) which is unaltered by a group of
transformations of the form x' = {ax + h)l{cx + d), finding, for your example, a region in
the plane of the complex variable x in which the function assumes every value just once.
Explain some general method of expressing automorphic functions when the group of
linear transformations is assigned.
202. In the equation
which is satisfied by the quarter periods K and iK' of the Jacobian elliptic functions
formed with the modulus k=\/z, shew that is a uniform function of the quotient
t = -^ of two solutions of the equation, being automorphic for the group generated by the
substitutions
(M + 2)and(^,^).
Shew how, by using this as an auxiliary differential equation, any linear differential
equation with uniform coefficients and three singular points can be solved in terms of
uniform functions.
GLOSSARY QF TECHNICAL TERMS.
{The numbers refer to the pages, ivhere the term occurs for the
first time in the book or is defined.)
Abbildung, conforme, 11.
Absolute convergence of series, 21 ; of pro-
ducts, 91.
Absoluter Betrag, 3.
Absolute value, 3.
Accidental singularity (pole), 17, 61.
Addition-theorem, algebraic, 344.
Adelphic order, 864.
Adjoint curves, 445.
Adjoint polynomials, 445.
Algebraic addition-theorem, 344.
Algebraic function, determined by an equation,
190.
Amplitude, 3.
Analytical curve, 458,' 478, 658.
Analytic function, 10; monogenic, 67.
Anharmonic group, 754.
Argument, 3.
Argument and parameter, interchange of, 513.
Arithmetic mean, method of the, 458.
Ausserwesentliche singuldre Stelle, 61.
Automorphic functions, 715, 753.
Betrag, absoluter, 3.
Bicursal, 555.
Bien defini, 190.
Bifacial surface, 372.
Birational transformation, 537.
Boundary, 369.
Branch, 16.
Branch-line, 385.
Branch-point, 17, 183.
Branch-section, 385.
Canonical resolution of surface, 402.
Categories of corners, cycles, 725, 729.
Circle, discriminating, 133.
Circle of convergence, 22.
Circuit, 374.
Class, or genus, (of connected surface), 371.
Class of doubly-periodic function of second
order, 263.
Class of equation, 395.
Class of group, 742.
Class of singularity, 177.
Class of tertiary-periodic function, 335.
Class of transcendental integral functions, 109.
Class-moduli, 545.
Combination of areas, 480.
Compound circuit, 374.
Conditional convergence of series, 21; of pro-
ducts, 91.
Conditional equation in Abel's theorem, 581.
Conformal representation, 11.
Conforme Abbildung, 11.
Congruent figures, 631, 724.
Conjugate edges, 725.
Connected surface, 359.
Connection, order of, 364.
Connectivity, 364.
Constant modulus for cross-cut, 427.
Contiguous regions, 724.
Continuation, 67.
Continuity, region of, 67.
Continuous substitution, 717.
Convergence of series, 22; of products, 91.
Convexity of normal polygon, 727.
Corner of region, 724.
Coupure, 165, 220.
Critical point, 17.
Cross-cut, 361.
Cross-line, 385.
Cycles of branches of algebraic function, 570.
Cycles of corners, 726.
830
GLOSSAEY OF TECHNICAL TERMS
Deficiency, 403.
Deformation of loop, 407.
Deformation of surface, 379.
Degree of cycle, 570.
Degree of pseudo-automorphic function, 785.
Degree of rational function on Biemann's
surface, 420.
Derivative, Schwarzian, 657.
•Dihedral group, 757.
Diramazione, punto di, 17.
Dirichlet's principle, 458.
Discontinuity, polar, 17.
Discontinuous groups, 717.
Discontinuous substitution, 717.
Discrete substitution, 717.
Discriminating circle, 133.
Divergence of series, 22 ; of products, 91.
Domain, 60.
Domaine, 60.
Dominant function, 39.
Double (or fixed) circle of elliptic substitution,
746.
Doubly-periodic function of first, second, third,
kind, 320, 321.
Edge of region, 724.
Edges of cross-cut, positive and negative, 424.
Eindndrig, 16.
Eindeutig, 16.
Einfach zusammenhcmgend, 360.
Element, 67.
Element of doubly-periodic function of third
kind, 338, 340.
Elementary integral of the second kind, third
kind, 446, 452.
Elliptic substitution, 631.
Equivalent homoperiodic functions, 260.
Essential singularity, 19, 61.
Exceptional value, 66.
Existence-theorem, 416, 455,
Factor, primary, 101.
Factorial functions, 531.
Families of groups, 740.
Finite groups, 719.
First kind, doubly-periodic function of the,
321.
First kind of Abelian integrals, 444.
Fixed (or double) points of substitution, 628.
Forlsetzung, 67.
Fractional factor for potential function, 476.
Fractional part of doubly-periodic function,
259.
Fuchsian functions, 753.
Fuchsian groups, 740.
Fundamental circle for group, 737.
Fundamental loops, 407.
Fundamental parallelogram, 237, 244.
Fundamental polyhedron (of reference for
space), 748.
Fundamental region (of reference for plane),
724.
Fundamental substitutions, 716.
Gattung (kind of integral), 444.
Genere, 109.
Genere (genus of connected surface), 371.
Genre (applied to singularity), 177.
Genre (applied to transcendental integral
functions), 109.
Genre (genus of connected surface), 371.
Genus (of connected surface), 371.
Genus (of equation), 395.
Genus (of group), 742.
Geschlecht, 395, 403.
Giramento, punto di, 17.
Gleichverzweigt, 419.
Grenze, natilrliche, 153.
Grenzkreis, 133.
Group of substitutions, 715.
Grmidzahl, 364.
Harmonic functions, 9.
Hauptkreis, 737.
Holomorphic, 17.
Homogeneous substitutions, 756.
nomographic transformation, or substitution,
625.
Homologous (points), 237.
Homoperiodic, 263.
Hyperbolic substitution, 631.,
Hyperellipti^ curves or equations, 565.
Improperly discontinuous groups, 718.
Index of substitution, 717.
Infinitesimal substitution, 636, 717.
Infinity, 17.
Integrals of the first kind, second kind, third
kind, Abelian, 444, 446, 452.
Interchange of argument and parameter, 513.
Invariants of elliptic functions, 295.
Inversion-problem, 517.
Irreducible circuit, 374.
Irreducible (point), 236, 237.
Isothermal, 707.
Kleinian functions, 753.
Kleinian group, 743.
GLOSSAEY OF TECHNICAL TERMS
831
Lacet, 182.
Lacunary functions, 166.
Level values, 269.
Ligne de passage, 385.
Limit, natural, 153.
Limitrophe, 724.
Linear cycles, 570.
Linear substitution, 625.
Loop, 182.
Loop-cut, 362.
Loxodromic substitution,
631.
Majorante, 39.
Mehrdeutig, 16.
Mehrfach zusainmenhdngend, 361.
Meromorphic, 17.
Modular-function, 767.
Modular group, 721.
Modulus, 3.
Modulus for cross-cut, constant, 427.
Modulus of periodicity (cross-cut), 427.
MonadeliDbic, 360.
Monodromic, 16.
Monogenic, 15.
Monogenic analytic function, 67.
Monotropic, 16.
Multiform, 16.
Multiple circuit, 374.
Multiple connection, 362.
Multiplicateurs, fonctions a, 531.
Multiplier of substitution, 628.
Natural limit, 153.
Natiirliche Grenze, 153.
Negative edge of cross-cut, 424.
Non-essential singularity, 61.
Normal (connected) surface, 381.
Normal form of linear substitution, 715.
Normal form of transformable equations, 567.
Normal function of first kind, second kind,
third kind, 508, 510, 511.
Normal polygon for substitutions, 728.
Order of a doubly-periodic function, 25?.
Order, of connection, adelphic, 364.
Order of rational function on Eiemann's
surface, 420.
Ordinary point, 60.
Origin of cycle, 570.
Orthomorphosis, 11.
Oscillating series, 21.
Parabolic substitution, 631.
Parallelogram, fundamental or primitive, 237,
244.
Path of integration, 21.
Period, 235.
Periodicity for cross-cut, modulus of, 427.
Permanent equation in Abel's theorem, 581.
Polar discontinuity, 17.
Pole, 17, 61.
Polyadelphic, 361.
Polyhedral functions, 706.
Poly tropic, 16.
Positive edge of cross-cut, 424.
Potential function, 457.
Primary factor, 101.
Primfunction, 101.
Primitive parallelogram, 244.
Products, convergence of, 91.
Properly discontinuous groups, 718.
Pseudo-periodicity, 301, 304, 320, 321.
Punto di diramazione, punto di giramento,
17.
Querschnitt, 361.
Eamification (of Eiemann's surface), 395.
Ramification, point de, 17.
Eational function, 84.
Eeal substitutions, 631.
Eeconcileable circuits, 374.
Eeducible circuit, 374.
Eeducible (point), 236, 237.
Eegion of continuity, 67.
Eegular, 17, 60.
Eegular singularities, 192.
Mepresentation conforme, 11.
Eesidue, 48.
Eesolution of surface, canonical, 402.
Retrosection, 362.
Eiemann's surface, 382.
Eoot, 17.
RUckkehrschnitt, 862.
Schleife, 182.
Schwarzian derivative, 657.
Second kind, doubly-periodic function of the,
821.
Second kind of Abelian integrals, 446.
Secondary-periodic functfcns, 322.
Section, 69, 165, 220.
Section (cross-cut), 361.
Series, convergence of, 22.
Sheet, 382.
Simple branch-points, 208.
Simple circuit, 374.
Simple connection, 360.
Simple curve, 24.
Simple cycle of loops, 408.
882
GLOSSARY OF TECHNICAL TERMS
Simple element for tertiary-periodic function,
338, 340.
Singular point, 17.
Singularity, accidental, 17, 61.
Singularity, essential, 19, 61.
Special function on Riemann's surface, 526.
Species of singularity, 177.
Sub-categories of cycles, 740.
Sub-rational representation of variables, 551.
Substitution, homogeneous, 756.
Substitution, linear or homographic, 625.
Syneetic, 17.
Umgebung, 60.
Unconditional convergence of series, 22 ; of
products, 91.
Unicursal, 548.
Unifacial surface, 372.
Uniform convergence of series, 22; of pro-
ducts, 91.
Uniform function, 16.
Verzweigungschnitt, 385.
Verzweigungspunkt, 17.
Taglio trasversale, 361.
Tertiary-periodic functions, 322.
Tetrahedral group, 759.
Thetafuchsian function, 776.
Third kind, doubly-periodic function of the, 321.
Third kind of Abelian integral, 452.
Transcendental function, 84.
Transformation, birational, 537.
Trasversale, 361.
Wesentliche singulare Stelle, 61.
Winding-point, 392.
Winding-surface, 392.
Windungspunkt, 17.
Zero, 17.
Zusammenhangend, einfach, mehrfach, 360,
361.
INDEX.
(The numbers refer to the pages.
Abel, 269, 518, 580.
Abel's formula for sum of transcendental
integrals, 585 : examples of, 587-590 ; ap-
plied to integrals of first kind, 590 ; of
second kind, 594; of third kind, 597.
Abel's Theorem on integrals, quoted, 519 ;
proved, 579-601 : the main result, 585.
Abelian transcendental functions, arising by
inversion of functions of the first kind on
a Eiemann's surface, 517;
Weierstrass's form of, 518.
Absolute convergence, of series, 21 ; of pro-
ducts, 91.
Accidental singularities, 17, 61, 78 ;
must be possessed by uniform function,
78;
form of function in vicinity of, 78;
are isolated points, 78;
number of, in an area, 82, 86;
if at infinity and there be no other
singularity, the function is polynomial,
83;
if there be a finite number of, and no
essential singularity, the uniform
function is rational and meromorphic,
■ 85.
Addition-theorem, for uneven doubly-periodic
function of second order and second class,
290;
for Weierstrass's ^-function, 307;
quasi-form of, for the (7-function and
tbe f-function, 307;
definition of algebraical, 344;
algebraical, is possessed by algebraical
functions, 344;
by simply- periodic functions, 345;
by doubly -periodic functions, 846;
function which possesses an algebraical, is
either (i) algebraical, 347;
or (ii) simply -periodic, 350, 352;
or (iii) doubly-periodic, 354;
F. F.
satisfies a differential equation be-
tween itself and its first derivative,
355;
condition that algebraical equation be-
tween three variables should express,
357;
form of, when function is uniform,
358.
Adjoint curves, 445.
Adjoint polynomial on Eiemann's surface,
quotient of one by another, is a special
function, 527.
Adjoint polynomials, 445.
Algebraic equation between three variables
should express an addition-theorem, condi-
tion that, 357;
Algebraic equation, defining algebraic multi-
form functions, 190 (see algebraic function);
genus of, 395 ;
for any uniform function of position on
a Eiemann's surface, 417.
Algebraic equation defines functions that are
analytic, 207.
Algebraic equation has roots, 88.
Algebraic function, cycle of branches of, 570:
birationally transformed, 571-577.
Algebraic function is analytic, 207.
Algebraic (multiform) functions defined by
algebraical equation, 190;
branch-points of, 191 ;
infinities of, are singularities of the
coefficients, 192 ;
graphical method for determination
of order of, 194 ;
branch-points of, 197;
cyclical arrangements of branches round
a branch-point, 200;
when all the branch-points are simple,
208;
in connection with Eiemann's surface,
386.
53
834
INDEX
Algebraic function on a Eiemann's surface,
integrals of, 436;
integrals of, everywhere finite, 438 ;
number of, in a special case, 438;
when all branch-points are simple, three
kinds of integrals of, 439 ;
infinities of integrals of, 440, 443;
branch-points of integrals of, 443.
Algebraic functions on a Eiemann's surface,
constructed from normal elementary func-
tions of second kind, 520;
smallest number of arbitrary infinities
to render this construction possible,
520;
Kiemann-Roch's theorem on, 521 ;
smallest number of infinities of, which,
f except at them, is everywhere uniform
and continuous, 523 ;
which arise as first derivatives of func-
tions of first kind, 524 ;
are infinite only at branch-points,
525;
number of infinities of, and zeros
of, 525;
most general form of, 526 ;
determined by finite zeros, 526;
Brill-Nother law of reciprocity for,
528;
determine a fundamental equation for a
given Eiemann's surface, 528;
relations between zeros and infinities of,
535.
Algebraic isothermal curves, families of, 707
et seq. (see isothermal curves).
Algebraic plane curve birationally transformed
into another with double points only, 569-578.
Algebraic relation between functions automor-
phic for the same infinite group, 788 ;
genus of, in general, 789.
Analytic function, monogenic, 67.
Analytic function represented by series of
polynomials, 69, 134.
Analytic function defined by algebraic equation,
207.
Analytical curve, 459, 478, 658;
represented on a circle, 478;
area bounded by, represented on a half-
plane, 658;
consecutive curve can be chosen at
will, 659.
Analytical test of a branch-point, 186.
Aiichor-ring conformally represented on plane,
612.
Anharmonic function, automorphic for the an-
harmonic group, 754.
Anharmonic group of linear substitutions, 754.
Anissimoff, 133.
Appell, 174, 223, 342, 343, 530, 531, 559, 570.
Appell's factorial functions, 531 (see factorial
functions).
Area, simply connected, can be represented
conformally upon a circle with unique cor-
respondence of points, by Eiemann's theorem,
654;
form of function for representation on a
plane, 657, 670; on a circle, 657;
bounded by analytical curve represented
on half- plane, 658 ;
bounded by cardioid on half-plane, 662 ;
of convex rectilinear polygon, 666 et
seq. (see rectilinear polygon) ;
bounded by circular arcs, 679 et seq. (see
curvilinear polygon).
Areas, combination of, in proof of existence-
theorem, 480.
Argand, 2.
Argument (or amplitude) of the variable, 3.
Argument of function possessing an addition-
theorem, forms of, for a value of the function,
347 et seq.
Argument and parameter of normal elementary
function of third kind, 515.
Ascoli, 459.
Automorphic function, 753 ;
constructed for infinite group in pseudo-
automorphic form, 771 et seq. (see
thetafuchsian functions) ;
expressed as quotient of two theta-
fuchsian functions, 784;
its essential singularities, 786 ;
number of irreducible zeros of, is the
same as the number of irreducible
accidental singularities, 786;
different, for same group are connected
by algebraical equation, 788; genus
of this algebraical equation in general,
789;
connection between, and general linear
differential equations of second order,
791;
modular-functions as examples of, 792.
Baker, 247, 396, 404, 519, 528, 530, 537, 567,
579; a rule for determining the genus of a
Eiemann's surface, 404.
Barnes, 103.
Barrier, impassable, in connected surface, 360 ;
can be used to classify connected sur-
faces, 361 ;
changed into a cut, 361.
INDEX
835
Beltrami, 658, 660, 661.
Bernoulli's numbers, 48.
Bertini, 570.
Bianchi, 751, 752.
Bicursal equations and curves, 555.
Biehler, 342.
Biermann, 67, 344.
Bifacial surfaces, 372, 380.
Birational transformation, 415, 587-579 ;
conserves genus of equation, 542 ;
conserves kind of function, 543 ;
conserves Sp-S + p class-moduli, 545.
Birational transformation of algebraic plane
curves, 569-578: of cycles of branches of
algebraic function, 571-577.
Birational transformation of equations of
genus zero, 550 ; of genus unity, 558 ; of
genus greater than unity, 566; of genus
greater than two, 569.
Blumenthal, 113.
Bolza, 716.
Bonnet, 611.
Bonola, 714.
Boole, 585.
Borchardt, 257.
Borda, 646.
Borel, 113, 134, 173.
Boundary of region of continuity of a function
is composed of the singularities of the
function, 68.
Boundary, defined, 369 ;
assigned to every connected surface, 361,
369;
edges acquired by cross-cut and loop-
cut, 362;
of simply connected surface is a single
line, 370;
effect of cross-cut on, 370;
and of loop-cut on, 371.
Boundary conditions for potential function,
460 (see potential function).
Boundary, functions on a Eiemann's surface
without, 491.
Boundary values of potential function for a
circle, 465;
may have limited number of finite dis-
continuities, 470;
include all the maxima and the minima
of a potential function, 476.
Boundaries of connected surface, relation
between number of, and connectivity,
371.
Branches of a function, defined, 16;
affected by branch-points, 180 et seq. ;
obtained by continuation, 180 ;
are uniform in continuous regions where
branch-points do not occur, 184;
which are affected by a branch-point,
can be arranged in cycles, 185;
restored after number of descriptions of
circuit round branch-point, 186 ;
analytical expression of, in vicinity of
branch-point, 187;
number of, considered, 188;
of an algebraic function, 190 (see alge-
braic function) ;
a function which has a limited number
of, is a root of an algebraic equation,
210.
Branch-lines, are mode of junction of the sheets
of Eiemann's surfaces, 385 ;
properties of, 386 et seq.;
free ends of, are branch-points, 386;
sequence along, how affected by branch-
points, 387;
system of, for a surface, 387;
special form of, for two-sheeted surface,
391;
when all branch-points are simple, 403 ;
number of, when branch-points are
simple, 412.
Branch-points, defined, 16, 183 ;
integral of a function round any curve
containing all the, 42;
effect of, on branches, 178, 180 et seq. ;
analytical test of, 186 ;
expression of branches of a function in
vicinity of, 187 ;
of algebraic functions, 191, 197 ;
simple, 208, 403;
number of simple, 209;
are free ends of branch-lines, 886;
effect of, on sequence of interchange
along branch-lines, 887 ;
joined by branch-lines when simple,
391;
deformation of circuit on Eiemann's
surface over, is impossible, 396 ;
circuits round two, are irreducible, 396 ;
number of, when simple, 402;
in connection with loops, 404 (see
loops) ;
canonical arrangement of, when simple,
411.
Brill, 404, 415, 528, 530, 570.
Brill-Nother law of reciprocity, 528.
Brioschi, 822, 828.
Briot, 531.
Briot and Bouquet, vi, 27, 44, 47, 197, 208,
246, 249, 257, 269, 519.
53—2
836
INDEX
Bromwich, 6, 21, 292.
Burnside (W.), 141, 402, 456, 638, 653, 664,
689, 716, 754, 774, 789.
Burnside (W. S.) and Panton, 440, 584.
Canonical form, of complete system of simple
loops, 409;
of Kiemann's surface, 413 ;
resolved, 414.
Canonical resolution of Eiemann's surface, 402.
Cantor, 176.
Cardioid, area bounded by, represented on strip
of plane, 662 ;
on a circle, 663.
Carslaw, 21, 23, 652, 714.
Casorati, 2, 27, 407.
Categories of corners, 725 (see corners).
Cathcart, 6, 20, 61.
Cauchy, v, vi, 24, 27, 31, 49, 50, 52, 59, 75,
82, 207, 214, 359, 585.
Cauchy's theorem on the integration of a
holomorphic function round a simple curve,
27;
and of a meromorphic function, 30;
on the expansion of a function in the
vicinity of an ordinary point, 50.
Cayley, 2, 11, 92, 397, 403, 552, 555, 557, 615,
622, 623, 629, 657, 658, 661, 665, 679, 705,
707, 710, 754, 756.
Cesaro, 113.
Chessin, 250.
Christoffel, 652, 666, 670, 679.
Chrystal, vi, 2, 6, 199, 218.
Circle, areas of curves represented on area of :
exterior of ellipse, 614;
interior of ellipse, 617 ;
interior of rectangle, 615, 674;
interior of square, 615, 674;
exterior of square, 674 ;
exterior of parabola, 618 ;
interior of parabola, 619 ;
half-plane, 619;
interior of semicircle, 620 ;
infinitely long strip of plane, 621 ;
any circle, by properly chosen linear
substitution, 627;
any simply connected area, by Eiemann's
theorem, 654;
interior of cardioid, 662 ;
interior of regular polygon, 678.
Circle of convergence of series, 22.
Circuits, round branch-point, effect of, on
branch of a function, 182, 184;
restore initial branch after . number of
descriptions, 186;
on connected surface, 374 ;
reducible, irreducible, simple, multiple,
compound, reconcileable, 374;
represented algebraically, 375 ;
drawn on a simply connected surface are
reducible, 376 ;
number in complete system for multiply
connected surface, 377;
cannot be deformed over a branch-point
on a Eiemann's surface, 397.
Circular functions obtained, by integrating
algebraical functions, 226;
on a Eiemann's surface, 430.
Class-moduli of equations under birational
transformation, 544: number of, 545.
Class, of transcendental integral function as
defined by its zeros, 109;
Laguerre's criterion of. 111;
simple function of given, 112 ;
essential singularity, 176 ;
tertiary-periodic function, positive, 335;
negative, 338;
(see genus] .
Classes of doubly-periodic functions of the
second order are two, 262.
Clebsch, 208, 247, 403, 407, 408, 411, 415, 453,
518, 519, 548, 554, 557, 569, 579.
Clifford, 380, 408.
Closed cycles of corners in normal polygon for
division of plane, 730 (see corners).
Combination of areas, in determination of
potential function, 480.
Complex variable defined, 1 ;
represented on a plane, 2 ;
and on Neumann's sphere, 4.
Compound circuits, 374.
Conditional convergence of series, 21; of pro-
ducts, 91.
Conditional equation in Abel's Theorem, 581.
Conditions that one complex variable be a
function of another, 7.
Conduction of heat, application of conformal
representation to, 649.
Conformal representation applied to hydro-
dynamics, 639 ; to electrostatics, 646 ; to
conduction of heat, 649.
Conformal representation of planes, established
by functional relation between variables, 11;
magnification in, 11 ;
used in Schwarz's proof of existence-
theorem, 478;
most general form of relation that secures,
is relation between complex variables,
606;
examples of, 614 et seq.
INDEX
837
Conformal representation of surfaces is secured
by relation between complex variables in the
most general manner, 606 ;
obtained by making one a plane, 607;
of surfaces of revolution on plane, 607 ;
of sphere on plane, 609 ;
Mercator's and stereographic projec-
tion, 609, 610;
of oblate spheroid, 612 ;
of ellipsoid, 612;
of anchor-ring, 612 ;
of surface of constant negative curvature,
613;
Eiemann's general theorem on, 654 ;
form of function for, on a plane, 657 ;
on a circle, 657.
Congruent regions by linear substitutions, 631,
724.
Conjugate edges of a region, 725 (see edges).
Conjugate functions, 9.
Connected surface, supposed to have aboundary,
360, 368, 375 ;
to be bifacial, 372 ;
divided into polygon s , Lhuilier ' s theorem
on, 372 ;
geometrical and physical deformation of,
379;
can be deformed into any other connected
surface of the same connectivity having
the same number of boundaries, if both
be bifacial, 380 ;
Klein's normal form of, 381;
associated with irreducible equation, 392.
Connection of surfaces, defined, 359 ;
simple, 360 ;
definition of, 362 ;
, multiple, 361 ;
definition of, 362 ;
affected by cross-cuts, 366 ;
by loop-cuts, 367 ;
and by slit, 368.
Connectivity, of surface defined, 364 ;
affected by cross-cuts, 366 ;
by loop-cuts, 367 ;
by sUt, 368 ;
of spherical surface with holes, 368 ;
in relation to irreducible circuits, 376 ;
of a Eiemann's surface, with one boun-
dary, 394;
with several boundaries, 396.
Constant, uniform function is, everywhere if
constant along a line or over an area, 72.
Constant difference of integral, at opposite
edges of cross-cut, 424 ;
how related for cross-cuts that meet, 425 ;
for canonical cross-cuts, 426 (see
moduli of periodicity).
Constant negative curvature, surfaces of, 712.
Construction of rational function on Eiemann's
surface, 519-524.
Contiguous regions, 724.
Continuation , of function by successive domains,
67;
Schwarz's symmetric, 70 ;
of function with essential singularities,
120;
of multiform function to obtain branches,
180.
Continuity of a function, region of (see region
of continuity).
Continuous convergence, 22.
Continuous group, 718.
Contour integration, 43-49.
Contraction of areas in conformal representa-
tion, 665.
Convergence of products, kinds of, 91.
Convergence, of series, kinds of, 22 ; circle of,
22 ; of products, 91.
Convex curve, area of, represented on half-
plane, deduced as the limit of the representa-
tion of a rectilinear polygon, 679.
Convex normal polygon for division of plane,
in connection with an infinite group, 728 ;
angles at corners of second category and
of third category, 730 ;
sum of angles at the corners in a cycle
of the first category is a submultiple
of four right angles, 731 ;
when given leads to group, 734 ;
changed into a closed surface, 742.
Corners, of regions, 724 ;
three categories of, for Fuchsian group,
725;
cycles of homologous, 726 ;
how obtained, 730 ;
' closed, and open, 730 ;
categories of cycles, 730 ;
of first category are fixed points of
elliptic substitutions, 734 ;
of second and third categories are fixed
points of parabolic substitutions, 734 ;
sub-categories of cycles of, 741 ;
open cycles of, do not occur in Kleinian
groups, 747.
Crescent changed into another of the same
angle by a linear substitution, 628 ;
represented on a half- plane, 684.
Criterion of character of singularity, 80 ;
class of transcendental integral function,
111.
838
INDEX
Critical integer, for expansion of a function in
an infinite series of functions, 148.
Cross-cuts, defined, 361 ;
effect of, on simply connected surface,
363;
on any surface, 363 ;
on connectivity of surface, 366 ;
on number of boundaries, 370 ;
and irreducible circuits, 377 ;
on Eiemann's surface, 398 ;
chosen for resolution of Eiemann's sur-
face, 399 ;
in canonical resolution of Eiemann's
surface, 401 ;
in resolution of Eiemann's surface in its
canonical form, 413 ;
difference of values of integral at opposite
edges of, is constant, 424 ;
moduli of periodicity for, 426 ;
number of independent moduli, 428 ;
introduced in proof of existence-
theorem, 487 et seq.
Curve, birational transformation of algebraic
plane, 569-578.
Curves, adjoint, 445.
Curvilinear polygon, bounded by circular arcs,
represented on the half-plane, 679 et seq. ;
function for representation of, 680 ;
equation which secures the representa-
tion of, 683 ;
connected with linear differential
equations, 684 ;
bounded by two arcs, 684 ;
bounded by three arcs, 685 (see curvi-
linear triangles).
Curvilinear triangles, equation for representa-
tion of, on half-plane, 685 ;
connected with solution of differential
equation for the hypergeometric series,
686;
when the orthogonal circle is real, 688 ;
any number of, obtained by inver-
sions, lie within the orthogonal
circle, 689 ;
equation is transcendental, 689 ;
discrimination of cases, 689, 690 ;
particular case when the three arcs
touch, 691 ;
when the orthogonal circle is imaginary,
692;
stereographic projection on sphere
so as to give spherical triangle
bounded by great circles, 693 ;
connected with division of spherical
surface by planes of symmetry of
inscribed regular solids, 694 et
seq.;
cases when the relation is algebraical
in both variables and uniform in
one, 694 ;
equations which establish the
representation in these cases,
697 et seq. ;
cases when the relation is algebraical
in both variables but uniform in
neither, 704 et seq.
Cycles of branches of algebraic function, 570:
birational transformation of, into linear
cycles, 571-577.
Cycles of corners, 726 (see corners).
Cyclical interchange of branches of a function
which are affected by a branch-point, 185 ;
when the function is algebraic, 200.
Darboux, 23, 53, 70, 83, 379, 613, 666, 679, 712.
Dedekind, 767, 771.
Deficiency of a curve, 403 ;
equal to genus of associated Eiemann's
surface, 403 ;
determined by Baker's rule, 404 ;
is an invariant for rational transforma-
tions, 415, 542.
Deformation, of a circuit on a Eiemann's surface
over branch-point impossible, 397 ;
of connected surfaces, geometrical and
physical, 379 ;
can be effected from one to another
if they be bifacial, be of the same
connectivity, and have the same
number of boundaries, 380 ;
to its canonical form of Eiemann's sur-
face with simple winding-points, 413 ;
of loops, 405 et seq. ;
of path of integration, of holomorphie
function does not affect value of the
integral, 30 ;
over pole of meromorphic function
affects value of the integral, 39 ;
of multiform function (see integral
of multiform function) ;
form of, adopted, 224 ;
effect of, when there are more
than two periods, 247 ;
on Eiemann's surface (see path of
integration) ;
of path of variable for multiform
functions, 181 ;
how far it can take place without
affecting the final branch, 181-
184.
INDEX
839
Deformation of surfaces of constant negative
curvature, 712.
Degree of a function on a Eiemann's surface,
420.
Degree of cycle of branches of algebraic
function, 570.
•De Haan, 47.
Derivative, Schwarzian, 657 (see Schwarzian
derivative).
Derivatives, a holomorphic function possesses
any number of, at points within its region,
36;
do not necessarily exist along the boun-
dary of the region of continuity, 36,
158;
superior limit for modulus of, 38 ;
of elliptic functions with regard to the
invariants, 311, 312.
Description of closed curve, positive and nega-
tive directions of, 3.
De Sparre, 114.
Differential equation of first order, satisfied by
uniform doubly-periodic functions, 277 ;.
in particular, by elliptic functions, 277,
278;
possessing uniform integrals, 283;
satisfied by function which possesses an alge-
braic addition-theorem, 356.
Differentiation of uniformly convei-ging
function-series, 156.
Dihedral function, automorphic for dihedral
group, 765 (see polyhedral functions).
Dihedral group, of rotations, 757 ;
of homogeneous substitutions, 758 ;
of linear substitutions, 759 ;
function automorphic for, 765.
Dingeldey, 381.
Dini, vi.
Directions of description of closed curve, 3.
Discontinuous, groups, 717 ;
properly and improperly, 718 ;
all finite groups are, 719 ;
division of plane associated with, 724
(see regions).
Discrete group, 717.
Discriminating circle for uniform function, 133.
Discrimination between accidental and essen-
tial singularities, 61, 80.
Discrimination of branches of a function ob-
tained by various paths of the variable, 181
-184.
Divergence, of series, 22 ; of products, 91.
Division of surface into polygons, Lhuilier's
theorem on, 372.
Dixon, 139, 589.
Domain of ordinary point, 60.
Dominant function, 39.
Double points of linear substitution, 628.
Double-pyramid, division of surface of cir-
cumscribed sphere by planes of symmetry,
694;
equation giving the conformal represen-
tation on a half-plane of each triangle
in the stereographic projection of the
divided spherical surface, 698.
Doubly-infinite system of zeros, transcendental
function having, 104.
Doubly-periodic functions, uniform, 235 ;
graphical representation of, 236 ;
those considered have only one essential
singularity which is at infinity, 257,
267, 281 ;
fundamental properties of uniform, 258
et seq.;
order of, 259 ;
equivalent, 260 ;
integral of, round parallelogram of
periods, is zero, 260 ;
sum of residues of, for parallelogram, is
zero, 262 ;
of first order do not exist, 262 ;
of second order consist of two classes,
262;
number of zeros equal to number of
infinities and of level points, 266 ;
sum of zeros congruent with the surfi of
the infinities and with the sum of the
level points, 267 ;
of second order, characteristic equation
of, 270 ;
zeros and infinities of derivative of,
271;
can be expressed in terms of any
assigned homoperiodic function
of the second order with an ap-
propriate argument, 273 ;
of any order with simple infinities can
be expressed in terms of homoperiodic
functions of the second order, 274 ;
are connected by an algebraical equation
if they have the same periods, 276 ;
differential equation of first order satis-
fied by, 276 ;
in particular, by elliptic functions,
277;
can be expressed rationally in terms of.
a homoperiodic function of the second
order and its first derivative, 279 ;
of second order, properties of (see second
order) ;
840
INDEX
Liouville's theorem as to, 281 ;
expressed in terms of the f- function, 302;
and of the (j-function, 305 ;
possesses algebraical addition-theorem,
344.
Doubly-periodic integral of differential equation
of first order, 283.
Du Bois-Eeymond, 158.
Durege, 64, 363, 381.
Dyck, 381, 716, 718.
Edges of cross-cut, positive and negative, 424,
499.
Edges of regions in division of plane by an
infinite group, 724 ;
two kinds of, for real groups, 725 ;
congruent, are of the same kind, 725 ;
conjugate, 725 ;
of first kind are even in number and can
be arranged in conjugate pairs, 726 ;
each pair of conjugate, implies a funda-
mental substitution, 726.
Eisenstein, 105, 107.
Electric force, electric intensity, 647.
Electrostatics, application of conformal repre-
sentation to, 646.
Elementary function of second kind, 509 (see
second kind of functions).
Elementary functions of third kind, 511 (see
third kind of functions).
Elementary integrals of second kind, 446 ;
determined by an infinity, except as to
additive integral of first kind, 448 ;
number of independent, 449 ;
connected with those of third kind, 453.
Elementary integrals of third kind, 452 ;
connected with integrals of second kind,
453;
number of independent, with same log-
arithmetic infinities, 453.
Elements of analytic function, 67 ;
can be derived from any one when the
function is uniform, 68 ;
any single one of the, is sufficient for
the construction of the function, 68.
Ellipse, area without, represented on a circle,
614;
area within, represented on a rectangle,
616;
and on a circle, 617.
•Ellipsoid conformally represented on plane,
612.
Elliptic equations, or curves, 555.
Elliptic functions and equations of genus unity,
556.
Elliptic functions, obtained by integrating mul-
tiform functions, in Jacobian form, 228 ;
in Weierstrassian form, 231, 293 et seq.;
■ on a Riemann's surface, 432 et seq.
Elliptic substitutions, 631, 633;
are either periodic or infinitesimal, 635;
occur in connection with cycles of cor-
ners, 741, 747.
Enneper, 771.
Equations, of genus greater than two, 566 :
normal form of, 569.
of genus two, 562-565: variables in,
expressible by sextic or quintic radical, 563:
only limited number of birational trans-
formations into one another, 566 : normal
form of, 567.
of genus unity, 554-562 : variables in,
expressible by quartic or cubic radical, 554,
and as uniform . elliptic functions, 556: bi-
rationally transformable into one another
with one arbitrary parameter, 558 : normal
form of, 567.
of genus zero, 548-554: variables in,
expressible as rational functions, 548 : bi-
rationally transformable into one another
with three arbitrary parameters, 550 : sub-
rational representation of variables in, 551,
made rational, 552 : normal form of, 567.
Equipotential lines in planar electrostatics, 647.
Equivalent homoperiodic functions, 260;
conditions of equivalence, 265.
Essential singularities, 19, 61 ;
uniform function must assume any value
at or near, 64, 116;
of transcendental integral function at
infinity, 90;
form of function in vicinity of, 118 ;
continuation of function possessing, 120 ;
form of function having finite number
of, as a sum, 121 ;
functions having unlimited number of,
Chap. VII. ;
line of, 165 ;
laeunary space of, 166 ;
classification of, into classes, 175 ;
into species, 177 ;
into wider groups, 177;
of pseudo-automorphic functions, 776;
of automorphic fjmctions, 786.
Essential singularities of groups, 637, 739;
are essential singularities of functions
automorphic for the group, 739 ;
lie on the fundamental circle, 739 ;
may be the whole of the fundamental
circle, 740.
INDEX
841
Exceptional values unattainable near an
essential singularity, 66.
Existence of functions on a Eiemann's surface
without boundary, 491.
Existence-theorem for functions on a given
Eiemann's surface, Chap. xvii. ;
methods of proof of, 459 ;
abstract of Schwarz's proof of, 460 ;
results of, relating to classes of functions
proved to exist under conditions, 496.
Expansion of a function in the vicinity of an
ordinary point, by Cauchy's theorem, 50 ;
within a ring, by Laurent's theorem, 54.
Expression of uniform function, in vicinity of
ordinary point, 50 ;
in vicinity of a zero, 75 ;
in vicinity of accidental singularity,
79;
in vicinity of essential singularity, 118 ;
having finite number of essential singu-
larities, as a sum, 122 ;
as a product when without acciden-
tal singularities and zeros, 125,
126;
as a product, with any number of
zeros and no accidental singu-
larities, 130;
as a product, with any number of
zeros and of accidental singulari-
ties, 132;
in the vicinity of any one of an infinite
number of essential singularities, 135 ;
having an assigned infinite number of
singularities over the plane, 137 ;
generalised, 138 ;
having infinity as its single essential
singularity, 140;
having unlimited singularities distrib-
uted over a finite circle, 140.
Expression of multiform function in the vicinity
of branch-point, 187.
Factor, generalising, of transcendental integral
function, 99;
primary, 101 ;
fractional, for potential-function, 476 ;
major and minor, 477.
Factorial functions, pseudo-periodic on a Eie-
mann's surface, 531 ;
their argument, 531;
constant factors (or multipliers) for cross-
cuts of, 532 ;
forms of, when cross-cuts are canon-
ical, 532;
general form of, 532 ;
expression of, in terms of normal ele-
mentary functions of the third kind,
533 et seq. ;
zeros and infinities of, 535 ;
cross-cut multipliers and an assigned
number of infinities determine a
limited number of independent, 537.
Factorial periodicity, 719.
Factors (or miTltipliers) of factorial functions
at cross-cuts, 532;
forms of, when cross-cuts are canonical,
532.
Falk, 239.
Famihes of groups, seven, 740 ;
for one set, the whole line conserved by
the group is a line of essential singu-
larity ; for the other set, only parts
of the conserved line are lines of
essential singularity, 741.
Finite groups of linear substitutions, 719, 754 ;
containing a single fundamental substi-
tution, 719;
anharmonic, containing two elliptic
fundamental substitutions, 720.
Finite number of essential singularities, func-
tion having, expressed as a sum, 122.
First kind of pseudo-periodic function, 320.
First kind of functions on a Eiemann's surface,
498;
moduli of periodicity of functions of,
500 et seq. ; '
relation between, and those of a
function of second kind, 503;
when the functions are normal, 508 ;
number of linearly independent functions
of, 505 ;
normal functions of, 508 ;
inversion of, leading to multiply periodic
functions, 515 ;
derivatives of, as algebraical functions,
524;
infinities and zeros of, 525 ;
conserved under birational transforma-
tion, 542.
First kind of integrals on Eiemann's surface,
444;
number of, linearly independent in par-
ticular case, 445 ;
are not uniform functions, 445 ;
general value of, 446 (see first kind of
functions) ;
sum of, expressed by Abel's Theorem,
590.
Fixed circle of elliptic Kleinian substitution,
when the equation is generalised, 747.
842
INDEX
Fixed points of linear substitution, 628.
Floquet, 329.
Form of argument for given value of function
possessing an addition-theorem, 347 et seq.
Fourier, 651.
Fractional factor for potential function, 476 ;
major, minor, 477.
Fractional part of doubly-periodic function,
259.
Fredholm, 54.
Fresnel's integrals, 44.
Fricke, vii, 153, 453, 523, 526, 530, 625, 704, 754.
Frobenius, 312, 322, 328.
Fuchs, 133, 771.
Fuchsian functions, 753 (see automorpMc
functions).
Fuchsian group, 723, 740 ;
if real, conserves axis of real quantities,
723;
when real, it is transformed by one
complex substitution and then con-
serves a circle, 737;
division of plane into two portions
within and without the fundamental
circle, 737;
families of, 740;
genus of, 742.
Function defined by algebraic equation is
analytic, 207.
Function on Eiemann's surface, construction
of rational, 523 ; special, 526.
Function, Eiemann's general definition of, 8 ;
relations between real and imaginary
parts of, 9 ;
equations satisfied by real and imaginary
parts of, 12 ;
monogenic, defined, 15 ;
uniform, multiform, defined, 16;
branch, and branch-point, defined, 16 ;
holomorphic, defined, 17;
meromorphic, defined, 17;
continuation of a, 67;
region of continuity of, 67 ;
element of, 67 ;
monogenic analytic, definition of, 67 ;
constant along a line or area, if uniform,
is constant everywhere, 73 ;
properties of uniform, without essential
singularities, Chap. iv. ;
rational integral, 84;
transcendental, 84;
having a finite number of branches is
a root of an algebraical equation,
210;
potential, 457 (see potential function).
Function possessing an algebraic addition-
theorem, is either algebraic, or algebraic
simply-periodic, or algebraic doubly-periodic,
347;
has only a finite number of values for
one value of the argument, 355 ;
if uniform, then either rational, or
simply-periodic, or doubly-periodic,
355 ;
satisfies a differential equation between
itself and its first derivative, 356.
Functional dependence of complex variables,
form of, adopted, 7;
analytical conditions for, 7;
establishes conformal representation,
11.
Functionality, monogenic, not coextensive with
arithmetical expression, 164.
Functions, expression in series of (see series of
functions).
Functions of two variables, Weierstrass's
theorem on regular, 203-6.
Fundamental circle of Fuchsian group, 737 ;
divides plane into two parts which are
inverses of each other with regard to
the circle, 738;
essential singularities of the group lie
on, 740.
Fundamental equation for a Biemann's surface
is determined by algebraical functions that
exist on the surface, 529.
Fundamental parallelogram for double period-
icity, 237, 244;
is not unique, 244.
Fundamental region (or polygon) for division
of plane associated with a discontinuous
group, 724;
can be taken so as to have edges of the
first kind cutting the conserved line
orthogonally, 728, 738;
in this case, called a normal polygon,
727;
which can be taken as convex,
728;
angles of, 730 (see convex normal
polygon) ;
characteristics of, 732.
Fundamental set of loops, 407.
Fundamental substitutions of a group, 716;
relations between, 717, 726, 732;
one for each pair of conjugate edges of
region, 726.
Fundamental systems of isothermal curves, 712 ;
given by a uniform algebraic function,
or a uniform simply -periodic function,
INDEX
843
or a uniform doubly-periodic function,
712;
all families of algebraic isothermal curves
are derived from, by algebraic equa-
tions, 713.
Galois, 715.
Gauss, 2, 11, 103, 458, 602, 607, 611, 712,
714.
General conditions for potential function, 460
(see potential function).
Generalised equations of Kleinian group, 745
(see Kleinian group) ;
polyhedral division of space in connec-
tion with, 747.
Generalising factor of transcendental integral
function, 99.
Genus of, algebraic equation associated with a
Kiemann's surface, 395 ;
between automorphic functions, 789 ;
connected surface, 371 ; conserved under
birational transformation, 542;
Fuchsian group, 742 ;
Eiemann's surface, 395;
of Eiemann's surface equal to deficiency
of associated curve, 403 ;
determined by Baker's rule, 404.
Genus zero, equations of, 548-554;
unity, equations of, 554-562;
two, equations of, 562-565 ;
curve of, transformable into a quar-
tic, 565.
Gordan, 208, 247, 407, 415, 453, 518, 519, 569,
579, 719.
Goursat, 82, 105, 172, 222, 223, 243, 342, 530,
676, 679, 752.
Graphical determination of, order of infinity of
an algebraic function, 194 ;
the leading term of a branch in the
vicinity of an ordinary point of the
coefficients of the equation, 196;
the branches of an algebraic function in
the vicinity of a branch-point, 199.
Graphical representation of periodicity of func-
tions, 236, 237.
Green, 458.
GreenhiU, 227.
Group of linear substitutions, 715 ;
fundamental substitutions of, 716 ;
relations between, 717 ;
continuous, and discontinuous (or dis-
crete), 717;
properly and improperly discontinuous,
718;
finite, 719 (see finite groups) ;
modular, with two fiindamental sub-
stitutions, 720 ;
division of plane into polygons
associated with, 721 et seq. ;
relation between the funda-
mental substitutions, 723 ;
division of plane for any discontinuous
group, 724 (see region) ;
fundamental region for, 724 ;
Fuchsian, 724, 740 (see Fuchsian group) ;
when real, conserves axis of real quanti-
ties, 724;
fundamental substitutions of, connected
with the pairs of conjugate edges of a
region, 726 ;
seven families of, 740 ;
conserved line in relation to the essential
singularities, 741 ;
Kleinian, 743 (see Kleinian group) ;
dihedral, 757;
tetrahedral, 759.
Grouping of branches of algebraical function
at a branch-point, 200.
Giinther, 530.
Guichard, 126, 176, 177, 256, 257.
Gutzmer, 53.
Gyld^n, 150.
Hadamard, 54, 113, 803.
Half-plane represented on a circle, 619;
on a semicircle, 620;
on an infinitely long strip, 621 ;
on a sector, 622;
on a rectilinear polygon, 665 et seq. (see
rectilinear polygon) ;
on a curvilinear polygon, bounded by
circular arcs, 79 et seq. (see curvilinear
polygon, curvilinear triangle).
Halphen, 105, 309, 312, 322, 332, 342, 343, 572.
Halphen's birational transformation of plane
curves, 572 ; used to transform cycle of
branches of algebraic function, 572-577.
Hankel, 153, 223.
Hardcastle, F., 381.
Hardy, 6.
Harnack, 6, 10, 20, 61, 459.
Heine, 223.
Helmholtz, 642, 646.
Henrici, 459.
Hermite, vii, 23, 48, 95, 103, 113, 134, 165,
219, 220, 222, 302, 322, 324, 326, 333, 342,
518, 531, 541, 547, 767, 771.
Hermite's sections for integrals of uniform
functions, 220.
Herz, 611.
844
INDEX
Hexagon, symmetrical about one diagonal,
area of, represented on half-plane, 678.
Hilbert, 134.
Hill, M. J. M., 69.
Hobson, 6, 21, 102.
Hodgkinson, 690.
Hofmann, 414.
Holder, 64.
Hole in surface, effect of making, on connec-
tivity, 367.
Holomorphie function, defined, 17;
integral of, round a simple curve, 27 ;
along a line, 28;
when line is deformed, 29 ;
when simple curve is deformed, 30 ;
has a derivative for points within, but
not necessarily on the boundary of,
its region, 36;
superior limit for modulus of derivatives
of, 38 ;
expansion of, in the domain of an ordi-
nary point, 50, 60;
within a -ring of convergence by
Laurent's theorem, 55.
Holzmiiller, 2, 391, 615, 625.
Homen, 172.
Homogeneous form of linear substitutions, 756.
Homogeneous substitutions, 756;
two derived from each linear substitu-
tion, 756;
dihedral group of, 758.
nomographic substitution connected with sphe-
rical rotation, 755.
nomographic transformation, or substitution,
625 (see linear substitution).
Homologous points, 237, 724.
Homoperiodic functions, 263;
when in a constant ratio, 263 ;
are connected by an algebraical equation,
263.
when equivalent, 265;
Hotiel, 2. .
Humbert, 519, 530.
Hurwitz, 456, 566, 718, 721, 771, 792.
Hydrodynamics, application of conformal repre-
sentation to, 639.
Hyperbolic substitutions, 631, 633;
neither periodic nor infinitesimal, 636;
do not occur in connection with cycles
of corners, 741, 748.
Hyperelliptic equations or curves, 565.
Hypergeometric series, solution of differential
equation for, connected with conformal repre-
sentation of curvilinear triangle, 685 et seq. ;
cases of algebraical solution,697 et seq.
Icosahedral (and dodecahedral) division of sur-
face of circumscribed sphere, 696;
equation giving the conformal represent-
ation on a half-plane of each triangle
in the stereographic projection of the
divided surface, 704.
Identical substitution, 716.
Imaginary parts of functions, how related to
real parts, 9 ;
equations satisfied by real and, 12.
Improperly discontinuous groups, 718 ;
example of, 749 et seq.
Index of a composite substitution, 716;
not entirely determinate, 717.
Infinite circle, integral of any function round,
41.
Infinite class of integral function, 113.
Infinitesimal curve, integral of any function
round, 40.
Infinitesimal substitution, 717.
Infinities, of a function defined, 17;
of algebraic function, 192.
Infinities of doubly-periodic functions, irre-
ducible, are in number equal to the irreducible
zeros, 266;
and, in sum, are congruent with their
sums, 267;
of pseudo-periodic functions (see second
kind, third kind).
Infinities of potential function on a Riemann's
surface, 495.
Integral function, 52;
of infinite class, 113.
Integral with complex variables, defined, 20;
elementary properties of, 22, 23 ;
over area changed into integral round
boundary, by Riemann's fundamental
lemma, 25;
of holomorphie function round simple
curve is zero, 28 ;
of holomorphie function along a line is
holomorphie, 29;
of meromorphic function round simple
curve containing one simple pole, 31;
round simple curve, containing seve-
ral simple poles, 33 ;
round curve containing multiple
pole, 37;
of any function round infinitesimal circle,
40;
round infinitely great circle, 41;
round any curve enclosing all the
branch-points, 42.
Integral of multiform function, between two
points is unaltered for deformation of path
INDEX
845
not crossing a branch-point or an infinity,
215;
round a curve containing branch-points
and infinities is unaltered when the
curve is deformed to loops, 216 ;
also when the curve is otherwise deformed
under conditions, 217;
round a small curve enclosing a branch-
point, 217;
round a loop, 224;
deformed path adopted for, 225 ;
with more periods than two, can be
made to assume any value by modi-
fying the path of integration between
the limits, 246.
Integral of uniform function round parallelo-
gram of periods, is zero when function is
doubly-periodic, 260;
general expression for, 261.
Integrals, at opposite edges of cross-cut, values
of, differ by a constant, 424;
when cross-cuts are canonical, 426 ;
discontinuities of, excluded on a Kie-
mann's surface, 427 ;
general value of, on a Eieraann's surface,
428;
of algebraic functions, 436 ;
when branch-points are simple, 438 ;
infinities of, of algebraic functions,
439;
first kind of, 444 ;
number of independent, of first kind,
445;
arenot uniform functionsof position,
445;
general value of, 446;
second kind of, 446 (see second kind) ;
elementary, of second kind, 446 (see
elementary integrals) ;
third kind of, 450 (see third kind) ;
elementary, of third kind, 452 (see
elementary integral) ;
connected with integrals of second
kind, 453.
Integration, Eiemann's fundamental lemma in,
24.
Interchange, cyclical, of branches of a function
affected by a branch-point, 185;
of algebraical function, 210.
Interchange of argument and parameter in
normal elementary function of the third
kind, 515.
Interchange, sequence of, along branch-lines
determined, 387.
Interchangeable substitutions, 719.
Invariants, derivatives of elliptic functions with
regard to the, 312 ;
as automorphic functions, 785.
Inversion-problem, 517;
of functions of the first kind with several
variables leading to multiply periodic
functions, 517 et seq.
Inversions at circles, even number of, lead to
lineo-linear relation between initial and final
points, 638.
Irreducible circuits, 374;
complete system contains same number
of, 375;
cannot be drawn on a simply connected
surface, 376;
round two branch-points, 398.
Irreducible equation and singleness of con-
nected surface, 392.
Irreducible, points, 236, 237, 724, 772;
zeros of doubly-periodic function are the
same in number as irreducible infini-
ties, 266;
hkewise the number of level-points,
266;
also of automorphic functions, 787;
sum of irreducible points is independent
of the value of the doubly-periodic
function, 267.
Isothermal curves, families of plane algebraical,
707;
form of equation that gives such families
as the conformal representation of
parallel straight lines, 710;
three fundamental systems of, 710 ;
all, are conformal representations of
fundamental systems by algebraical
equations, 711;
isolated, may be algebraical by other
relations, 711.
Isothermal lines in conduction of heat, 650.
Jacobi, 108, 223, 228, 239 et seq., 278, 518,
592, 611, 612.
Jacobi's theorem in algebraic equations used to
deduce Abel's Theorem, 592-594.
Jeans, 649.
Jordan, 40, 87, 222, 570, 716.
Kapteyn, 740.
Kinds of edges in region for Fnchsian group,
725 (see edges).
Kinds of pseudo-periodic functions, three prin-
cipal, 320, 321 ;
examples of other, 342.
Kirchhoff, 628, 641. .
846
INDEX
Klein, vii, 153, 381, 417, 453, 458, 518, 523,
526, 530, 545, 566, 579, 612, 625, 631, 679,
704, 716 et seq., 753 et seq.
Kleinian functions, 753 (see automorphic
functions).
Kleinian group, 748;
conserves no fundamental line, 743;
generalised equations of, applied to space,
745;
conserve the plane of the complex
variable, 745 ;
double (or fised) circle of elliptic
substitution of, 746 ;
polygonal division of plane by, 746;
polyhedral division of space in connec-
tion with generalised equations of,
747;
relation between polygonal division of
plane and polyhedral division of space
associated with, 748.
Konigsberger, 269, 519.
Kopeke, 161.
Korkine, 608, 611.
Krause, 342.
Krazer, 519.
Kronecker, 153.
Lachlan, 688, 692.
Lacunary functions, 166.
Lagrange, 608, 611.
Laguerre, 109, HI, 112.
Laguerre's criterion of class of transcendental
integral function. 111.
Lamb, 646.
Lame, 328, 707.
Lamp's differential equation, 328 ;
can be integrated by secondary periodic
functions, 330;
general solution for integer value of n,
331;
special cases, 332.
Laurent, 50, 54, 57, 58, 82, 252, 253.
Laurent's theorem on the expansion of a func-
tion which converges within a ring, 54.
Law of reciprocity, Brill-Nother's, 528.
Leading term of a branch in vicinity of an
ordinary point of the coefficients of the
equation determined, 196.
Leathern, 646.
Legendre, 228.
Lerch, 161.
Level places are isolated points, 74.
Lhuilier, 372.
Lhuilier's theorem on division of connected
surface into polygons, 372.
Limit, natural, of a power-series, 153.
Lindelof, 49.
Lindemann, 403, 530.
Linear cycles of branches of algebraic func-
tions, 570 ; all cycles can be birationally
transformed into, 577.
Linear differential equations of the second
order, connected with automorphic functions,
791.
Linear substitution, 625;
equivalent to two translations, a reflexion
and an inversion, 626;
changes straight lines and circles into
circles in general, 627 ;
can be chosen so as to transform any
circle into any other circle, 628;
changes a plane crescent into another of
the same angle, 628;
fixed points of, 628;
multiplier of, 628 ;
condition of periodicity, 629;
parabolic, 631 ;
and real, 632;
elliptic, 631;
and real, 633 ;
is either periodic or infinitesimal,
635;
hyperbolic, 631 ;
and real, 633 ;
loxodromie, 631, 635 ;
can be obtained by any number of pairs
of inversions at circles, 637;
group of, 715 et seq. (see group) ;
normal form of, 715 ;
identical, 716 ;
algebraical symbols to represent, 716 ;
index of composite, 716 ;
infinitesimal, 717;
interchangeable, 719;
in homogeneous form, 756.
Lines of flow in conduction of heat, 649,
650.
Liouville, 190, 249, 257, 269.
Liouville's theorem on doubly-periodic func-
tions, 281.
Lippich, 363, 381.
Logarithmic differentiation of converging
products is possible, 92.
Logarithmic infinities, integral of third kind
on a Eiemann's surface must possess at
least two, 452.
Loop-cuts, defined, 362 ;
changed into a cross-cut, 367;
effect of, on connectivity, 367;
on number of boundaries, 371.
INDEX
847
Loops, defined, 182;
effect of a loop on a branch, is unique,
184;
symbol to represent effect of, 405;
change of, when loop is deformed,
406;
fundamental set of, 407 ; '
simple cycle of, 408;
canonical form of complete system of
simple, 409.
Love, 672.
Loxodromic substitutions, 631, 635;
neither periodic nor infinitesimal, 637 ;
do not occur in connection with cycles
of corners, 747.
Liiroth, 407, 408, 551, 554.
Magnification in conformal representation, 11,
603;
in star-maps, 611.
Mair, 381.
Major fractional factor, 477.
Maps, 611.
Mathews, 751.
Mathieu, 652.
Maximum and minimum values of potential
function for a region he on its boundary,
476.
Maxwell, 458.
Mercator's projection of sphere, 610.
Meromorphic function, defined, 17;
integral unchanged by deformation of
simple curve in part of plane where
function is uniform, 31;
integral round a simple curve, containing
one simple pole, 31 ;
round a curve containing several
simple poles, 33 ;
round a curve containing multiple
pole, 36;
cannot, without change, be deformed
across pole, 39;
is form of uniform function with a
limited number of accidental singu-
larities, 85 ;
all singularities of rational, are acci-
dental, 87.
Meyer, 707.
Michell, 670.
Minding, 712.
Minimum number of integrals in terms of
which any number is expressible by Abel's
Theorem, 599 ; the same as genus of equa-
tion, 599.
Minor fractional factor, 477.
Mittag-Leffler, vi, 68, 69, 70, 134 et seq.,
176, 322, 324, 326. '
Mittag-Leffler' s theorem on the expression of a
uniform function over its whole region of
existence, 69.
Mittag-Leffler's theorems on functions having
an unlimited number of singularities, dis-
tributed over the whole plane, 134;
distributed over a finite circle, 183.
Mobius, 372, 623, 704.
Modular-function defined, 767;
connected with elliptic quarter-periods,
767;
(see modular group) ;
as automorphic function, 792.
Modular group of substitutions, 719 ;
is improperly discontinuous for real
variables, 718 ;
division of plane into polygons, asso-
ciated with, 720 et seq. ;
relation between the fundamental sub-
stitutions of, 723;
for modulus of elliptic integral, 768;
for the absolute invariant of an elliptic
function, 770.
Moduh of periodicity, for cross-cuts, 427 ;
values of, for canonical cross-cuts, 427 ;
namber of linearly independent on a
surface, 428;
examples of, 429 et seq. ;
introduced in proof of existence-theorem ,
487 et seq. ;
of function of first kind on a Riemann's
surface, 498 et seq. ;
relation between, of a function of first
kind and a function of second kind, 503 ;
properties of, for normal function of
first kind, 508;
of normal elementary function of second
kind are algebraic functions of its
infinity, 510;
of normal elementary function of third
kind are expressed as normal functions
of first kind of its two infinities, 513.
Modulus of variable, 3.
Monogenic, defined, 15;
function has any number of derivatives,
36;
analytic function, 67.
Monogenic functionality not coextensive with
arithmetical expression, 164.
Multiform function, defined, 16;
elements of, in continuation, 68;
expression of, in vicinity of a branch-
point, 187;
848
INDEX
defined by algebraic equation, 190 (see
algebraic function) ;
integral of (see integral of multiform
function) ;
is uniform on Eiemann's surface, 384,
390.
Multiple circuits, 374.
Multiple periodicity, 247 ;
of uniform function of several variables,
248.
Multiplication-theorem, 344.
Multiplicity of zero, 75 ;
of pole, 80;
of a function on a Eiemann's surface,
421.
Multiplier of linear substitution, 628.
Multipliers of factorial functions at cross-cuts,
532;
forms of, when cross-cuts are canonical,
532.-
Multiply connected surface, 'SSO;
defined, 860;
connectivity modified by cross-cuts, 364 ;
by loop-cuts, 367;
and by slit, 368;
boundaries of, affected by cross-cuts, 370 ;
relation between boundaries of, and con-
nectivity, 371 ;
divided into polygons, Lhuilier's theorem
on, 372;
number of circuits in complete system
of circuits on, 377.
Multiply-periodic uniform functions of n vari-
ables, cannot have more than 2n periods.
248;
obtained by inversion of functions of
first kind, 515 et seq.
Natural limit, of a power-series, 153;
of part of plane, 689 ;
for pseudo-automorphic function with
certain families of groups, 777.
Negative curvature, surfaces of constant, 712.
Negative edge of cross-cut, 424, 499.
Nekrassoff, 133.
Netto, 716.
Neumann, vii, 5, 6, 42, 182, 190, 363 et seq.,
384, 401, 458, 459, 518, 531, 535, 586.
Neumann's sphere used to represent the vari-
able, 4;
used for multiform functions, 182.
Normal elementary function of second kind,
509 (see second kind of functions).
Normal elementary function of third kind, 510
(see third kind of functions).
Normal form of equations subject to birational
transformation, 567-569.
Normal form of linear substitution, 715.
Normal functions of first kind, 508 (see first
kind of functions).
Normal polygon for division of plane, 728;
can be taken convex, 728 (see convex
normal polygon).
Normal surface, Klein's, as a surface of refer-
ence of given connectivity and number of
boundaries, 381, 413.
Nother, 404, 528, 530, 570.
Number of zeros of uniform function in any
area, 75, 77, 82, 86;
of periodic functions (see doubly-periodic
functions, second kind, third kind) ;
of pseudo-automorphic functions (see
pseudo-automorphic functions).
Octahedral (and cubic) division of surface of
circumscribed sphere, 695 ;
equation giving the conformal repre-
sentation on a half-plane of each
triangle in the stereographic projec-
tion of the divided surface, 701.
Open cycles of corners in normal polygon for
division of plane by Fuchsian group, 710
(see corners);
do not occur in division of plane by
Kleinian group, 747.
Order (Borel's) of integral function, 113.
Order of a function on a Eiemann's surface,
420.
Order of doubly-periodic function, 259.
Order of infinity of a multiform function deter-
mined, 193.
Ordinary point of a function, 60;
domain of, 60.
Origin of cycle of branches of algebraic
function, 570.
Oscillating series, 21.
Painleve, 70, 134, 165.
Parabola, area without, represented on a circle,
618;
area within, represented on a circle, 619.
Parabolic substitutions, 631, 632;
neither periodic nor infinitesimal, 636 ;
occur in connection with cycles of cor-
ners, 741, 747.
Parallelogram for double periodicity, funda-
mental, 238, 243;
edges and corners in relation to zeros
and to accidental singularities of func-
tions, 258.
, INDEX
849
Parametric integer of thetafuchsian functions,
784.
Path of integration, 20 ;
can be deformed in region of holomor-
phic function without affecting the
value of the integral, 30 ;
on a Eiemann's surface, can be de-
formed except over a discontinuity,
422.
Periodic hnear substitutions, 629 ;
are elliptic, 633.
Periodicity of uniform functions, of one variable,
235 et seq. ;
of several variables, 247.
Periodicity, modulus of, 427 (see moduli of
periodicity).
Periods of a function of one variable, 235 ;
cannot have a real ratio when the func-
tion is uniform, 237;
cannot exceed two in number indepen-
dent of one another if function be
uniform, 242.
Permanent equation in Abel's Theorem, 581.
Phragmen, 134, 444, 457, 666.
Picard, 64, 66, 166, 329, 343, 491, 530, 560,
566, 569, 751.
Pincherle, 174, 719.
Plane used to represent variation of complex
variable, 2.
Pochhammer, 222.
Poincare, vii, 39, 113, 114, 166, 172, 342, 344,
566, 623, 632 et seq., 637, 716 et seq., 740,
752 et seq.
Poisson, 458.
Poles of a function defined, 17, 61.
Polyhedral division of space in connection with
generalised equations of group of Kleinian
substitutions, 748.
Polyhedral functions, connected with conformal
representation, 696 et seq. ;
for double-pyramid, 697, 766 ;
for tetrahedron, 698-764 ;
for octahedron and cube, 700 ;
for icosahedron and dodecahedron, 703.
Polynomials, adjoint, 445.
Polynomials, analytic function represented by
series of, 70, 134.
Polynomials on a Eiemann's surface, adjoint,
lead to special functions, 527.
Position on Eiemann's surface, most general
uniform function of, 417;
their algebraical expression, 419 ;
has as many zeros as infinities, 420.
Positive edge of cross-cut, 424, 459.
Potential function, defined, 457;
F. F.
conditions satisfied by, when derived
from a function of position on a Eie-
mann's surface, 467;
general conditions assigned to, 460;
boundary conditions assigned to, 460;
Green's integral- theorems connected
with, 461 et seq. ;
is uniquely determined for a circle by
general conditions and continuous
finite boundary values, 463 ;
integral expression obtained for,
satisfies the conditions, 467 ;
the boundary values for circle may
have finite discontinuities at a
limited number of isolated points,
470;
properties of, for a circle, 475 ;
maximum and minimum values of, in a
region, lie on the boundary, 476 ;
is determined by general conditions and
boundary values, for area conformally
representable on area of a circle, 478 ;
for combination of areas when it
can be obtained for each sepa-
rately, 480;
for area containing a winding-point,
485;
for any simply connected surface,
486;
introduction of cross-cut moduli for, on
a doubly connected surface, 487 ;
on a triply connected surface, 490 ;
on any multiply connected surface,
491;
number of linearly independent, every-
where finite, 495, 505 ;
introduction of assigned infinities, 495 ;
classes of, determined, 496 ;
classes of complex functions derived from,
with the respective conditions, 496.
Power-series, as elements of an analytical
function, 67 et seq., 152 et seq.;
region of continuity of, consists of one
connected part, 152 ;
may have a natural limit, 153.
Primary factor, 101.
Primitive parallelogram of periods, 244.
Pringsheim, 21, 91, 162, 289.
Product-form of transcendental integral func-
tion with infinite number of zeros over whole
plane, 99.
Products, convergence of, 91.
Prym, 400, 401, 417, 459, 519, 531.
Pseudo-automorphic functions, 777 (see theta-
fuchsian functions).
54
850
INDEX
Pseudo-periodic functions, Chap, xii.;
of the first kind, 320 ;
of the second kind, 321 ;
properties of (see second kind) ;
of the third kind, 321 ;
properties of (see third kind) ;
on a Kiemann's surface (see factorial
functions) .
Pseudo-periodicity of the ^-function, 301 ;
of the (T-function, 305.
Puiseux, 197.
Quadrilateral, area of, represented on half-
plane, 676;
determination of fourth angular point,
three being arbitrarily assigned, 678.
Quartic transformable into sextic curve, 546 ;
into another quartic, 547.
EafEy, 896.
Eamification of a Eiemann's surface, 395.
Eatio of periods of uniform periodic function
cannot be real, 238.
Eational function on Eiemann's surface, how
to construct, 523.
Eational integral of differential equation of
first order, 283.
Eational representation of variables in equation
of genus zero, 548.
Eational transformation, 537, 579.
Eauseuberger, 342, 719.
Eeal and imaginary parts of functions, how
related, 9;
equations satisfied by, 12 ;
each can be deduced from the other, 12.
Eeal potential function, 457 (see potential
function).
Eeal substitutions, 723 (see Fuchsian group).
Eeciprocity, Brill-Nother's law of, 528.
Eeconcileable circuits, 374.
Eectangle, area within, represented on a circle,
613;
and on an ellipse, 615 ;
on a half-plane, 674, 675.
Eectilinear polygon, convex, represented on
half- plane, 666 et seq.;
function for representation of, 668 ;
equation which secures the representa-
tion of, 668 ;
three angular points (but not more) may
be arbitrarily assigned in the repre-
sentation, 670 ;
determination of fourth for quadri-
lateral, 677;
three sides, 673 (see triangle) ;
four sides, 674 (see rectangle, squa,re
quadrilateral) ;
limit in the form of a convex curve, 678.
Eeducible circuits, 374.
Eeducible points, 236, 237.
Eegion of continuity, of a uniform function,
67, 150;
bounded by the singularities, 68 ;
of a power-series consists of one con-
nected part, 152;
may have a natural limit, 153 ;
of a series of uniform functions, 153 et
seq.;
of multiform function, 179.
Eegions in division of plane associated with
discontinuous group:
fundamental, 724;
uniform correspondence between, 724 ;
contiguous, 724;
edges of, 724 (see edges) ;
corners of, 724 (see corners).
Eegular functions of two variables, Weier-
strass's theorem on, 204-6.
Eegular in vicinity of ordinary point, function
is, 60.
Eegular polygon, area of, conformally repre-
sented on a circle, 678.
Eegular singularities of algebraical functions,
192.
Eegular solids, planes of symmetry of, dividing
the surface of the circumscribed sphere, 694
et seq.
Eepresentation, conformal, 11 (see conformal
representation).
Eepresentation of complex variable on a plane,
2;
and on Neumann's sphere, 4.
Eesidue of function, defined, 48;
when the function is doubly-periodic, the
sum of its residues is zero, 261.
Eesidues (Cauchy's) in Abel's Theorem, 585.
Eesolution of Eiemann's surface, 398 et seq.;
how to choose cross-cuts for, 399 ;
canonical, 402;
when in its canonical form, 413.
Eevolution, surface of, conformally represented
on a plane, 608.
Eiemann, v, vi, vii, 8, 10, 15, 24, 158, 214, 220,
359 et seq., 372 et seq., 416 et seq., 421,
453, 458, 459, 509, 518, 521, 527, 530, 543,
545, 548, 567, 611, 654, 792.
Eiemann, J., 459.
Eiemann-Eoch's theorem on algebraic functions
having assigned infinities, 521; comple-
mented by Brill-Nother law, 528.
INDEX
851
Riemann's definition of function, 8.
Riemann's fundamental lemma in integration,
24.
Riemann's surface, aggregate of plane sheets,
382;
used to represent algebraic functions, 384 ;
sheets of, joined along branch-lines, 385 ;
can be taken in spherical form, 393;
connectivity of, with one boundary, 394 ;
with several boundaries, 396 ;
genus of, 395 ;
ramification of, 395;
irreducible circuits on, 397;
resolution of, by cross-cuts into a simply
connected surface, 398 at seq. ;
canonical resolution of, 402;
form of, when branch-points are simple,
411;
* deformation to canonical form of,
412;
resolution of, in canonical form, 414;
uniform functions of position on, 417 ;
their expression and the equation
satisfied by them, 419;
have as many zeros as infinities,
420;
integrals of algebraic functions on a, 423
et seq.;
existence-theorem for functions on a
given, 455 ;
functions on (see first kind, second kind,
third kind of functions, algebraic
functions).
Riemann's theorem on conformalrepresentation
of any plane area, simply connected, on area
of a circle, 654.
Ritter, 754.
Roch, 521, 530.
Roots of a function, defined, 17 ; of an algebraic
equation, 88.
Rotations, connected with linear substitutions,
754;
groups of for regular solids, 757 ;
dihedral group of, 757 ;
tetrahedral group of, 759.
Rouche, 53.
Rowe, 580.
Runge, 134.
Salmon, 403, 415, 524.
Schlafli, 666.
Schlesinger, 754.
Schlomilch, 2.
Schonflies, 679, 752.
Schottky, 653, 753.
Schroder, 162.
Schwarz, vii, 13, 70, 161, 344, 455 et seq., 491,
566, 617, 619, 654 et seq.
Schwarz- Christoff el transformation, 670.
Schwarz's symmetric continuation, 70.
Schwarzian derivative, used in conformal re-
presentation, 657, 680 et seq.
Scott, C. A., 578.
Second kind of pseudo-periodic function, 321 ;
Hermite's expression for, 324, 326;
limiting form of, when function is
periodic of the first kind, 325,
327;
Mittag-Leffler's expression for, in inter-
mediate case, 325, 327 ;
number of irreducible infinities same
as the number of irreducible zeros,
327;
difi'erence between the sum of irreducible
infinities and sum of irreducible zeros,
328;
expressed in terms of the cr-fnnction,
328;
used to solve Lamp's differential equa-
tion, 328.
Second order of doubly-periodic functions (see
also doubly-periodic functions), properties of.
Chap. XI. ;
of second class and odd, 286 ;
connected with Jacobian elliptic
functions, 289;
addition-theorem for, 290 ;
of first class and even, illustrated by
Weierstrassian elliptic functions, 293
et seq. ;
of second class and even, -313 et seq.
Second kind, of functions on a Riemann's sur-
face, 498 ;
relation between moduh of periodicity of
functions of, and those of a function
of first kind, 503;
elementary function of, is determined by
its infinity and moduli, 509 ;
normal elementary function of, 509 ;
moduli of periodicity of, 510 ;
used to construct algebraic functions
on a Riemann's surface, 520.
Second kind, of integrals on a Riemann's sur-
face, 446;
elementary integrals of, 446 ;
general value of, 448 ;
elementary integrals of, determined by
an infinity except as to integral of
first kind, 448 ;
number of, 449 ;
852
INDEX
(see second kind of functions) ;
sum of, expressed by Abel's Theorem, 594.
Secondary periodic function, 322 (see second
kind of pseudo-periodic function).
Sections for integrals of uniform functions,
Hermite's, 69, 165, 220.
Sector on a half-plane, 622.
Seidel, 162.
Semicircle represented on a half-plane, 620;
on a circle, 620.
Sequence of interchange along branch-lines
determined, 387.
Series, convergence of, 21.
Series of functions, expansion in, 185 et seq. ;
region of continuity of, 156 ;
represents the same function throughout
any connected part of its region of
continuity, 157;
may represent different functions in dis-
tinct parts of its region of continuity,
162.
Series of polynomials representing analytic
function, 69, 134.
Series of powers, expansion in, 50 et seq. ;
function determined by, is the same
throughout its region of continuity,
152;
natural limit of, 153.
Serret, 716.
Sextic hyperelliptic curve transformable into
quartic, conditions, 546.
Sheets of a Eiemann's surface, 382 ;
relation between variable and, 384 ;
joined along branch-lines, 385.
Siebeck, 615, 710.
Simple branch-points for algebraic function,
208;
number of, 209, 403 ;
in connection with loops, 404;
canonical arrangement of, 411.
Simple circuit, 374.
Simple curve, defined, 24 ;
used as boundary, 369.
Simple cycles of loops, 408 ;
number of independent, 409.
Simple element for tertiary periodic functions,
of positive class, 338 ;
of negative class, 340.
Simply connected surface, 360 ;
defined, 362;
effect of cross-cut on, 363 ;
and of loop-cut on, 367;
circuits drawn on, are reducible, 376 ;
winding-surface containing one winding-
poiilt is, 395.
Simply infinite system of zeros, function having,
101.
Simply-periodic functions, 237 ;
graphical representation, 237, 251;
properties of, with an essential singu-
larity at infinity, 252 et seq. ;
when uniform, can be expressed as series
of powers of an exponential, 253;
of most elementary form, 255 ;
limited class of, considered, 257;
possess algebraical addition-theorem,
345.
Simply-periodic integral of differential equa-
tion of first order, 283.
Single connected surface associated with irre-
ducible equation or with repetition of
irreducible equation, 393.
Singular line, 165.
Singular points, 17. *
Singularities, accidental, 17 (see accidental
singularity) ;
essential, 19 (see essential singularity) ;
discrimination between, 61, 80;
• bound the region of continuity of the
function, 67;
must be possessed by uniform functions,
78;
of algebraical functions, regular, 192.
Singularity of a coefficient of an algebraic equa-
tion is an infinity of a branch of the function,
193.
Slit, effect of, on connectivity of surface, 368.
Special function on Eiemann's surface, 508 ;
is quotient of one adjoint polynomial by
another, 517.
Species of essential singularity, 177.
Sphere conformally represented on a plane,
609;
Mercator's projection, 609;
stereographic projection, 610.
Spherical form of Eiemann's surface, 393 ;
related to plane form, 393.
Spherical surface with holes, connectivity of,
368.
Spheroid, oblate, conformally represented on
plane, 612.
Square, area within, represented on a circle,
615, 674;
on a half-plane, 673, 675;
area without, represented on a circle, 674.
Stahl, 519, 753, 788.
Star-shaped region of continuity, constructed
by Mittag-Leffler, 69.
Stereographic projection of sphere on plane as
a conformal representation, 610;
INDEX
853
of curvilinear triangle on the surface of
a sphere, 693.
Stickelberger, 312, 519.
Stieltjes, 172, 173.
Stokes, 458.
Stolz, vi.
Straight line changed into a circle by a linear
substitution, 626.
Stream-lines, 640, 641 et seq.
Strip of plane, infinitely long, represented on
half -plane, 621;
on a circle, 619;
on a cardioid, 662.
Subcategories of cycles of corners, 741.
Sub-rational representation of variables in
equation of genus zero, 551.
Substitution, linear or homographic, 625 (see
linear substitution).
Sum of residues of doubly-periodic function,
relative to a fundamental parallelogram, is
zero, 261.
Sum of transcendental integrals, Abel's expres-
sion for, 583 : examples of, 587-590 : of first
kind, 590: of second kind, 594 : of third kind,
597 : minimum number equivalent to, 599.
Surface, connected, 359 ;
has a boundary assigned, 360, 368, 375;
effect of any number of cross-cuts on, 363;
connectivity of, 364;
affected by cross-cuts, 366;
by loop-cuts, 367 ;
and by slit, 368 ;
genus of, 371 ;
of constant negative curvature repre-
sented on a plane, 613, 712 ;
supposed bifacial, not unifacial, 872;
Lhuilier's theorem on division of, into
polygons, 372;
Biemann's (see Eiemann's surface).
Symbol for loop, 405 ;
change of, when loop is deformed, 406.
Symmetric continuation, Schwarz's, 70.
System of branch-lines for a Eiemann's surface,
387.
System of zeros for transcendental function,
simply-infinite, 101;
doubly-infinite, 104;
cannot be triply-infinite arithmetical
series, 108;
used to define its class, 109.
Tannery, vi, 162.
Tannery's series of functions representing dif-
ferent functions in distinct parts of its region
of continuity, 162.
Teixeira, 174.
Tertiary periodic functions, 322 (see third kind).
Test, analytical, of a branch-point, 186.
Tetrahedral division of surface of circumscribed
sphere, 695;
equation giving the conformal represent-
ation on a half-plane of each triangle
in the stereographic projection of the
divided surface, 699.
Tetrahedral function, automorphie for tetra-
hedral group, 764 (see polyhedral functions).
Tetrahedral group, of rotations, 759 ;
of substitutions, 761 ;
in another form, 762 ;
function automorphie for, 764.
Thetafuchsian functions, 776;
their essential singularities, 776 ;
exist either only within the fundamen-
tal circle, or over whole plane, accord-
ing to family of group, 777;
pseudo-automorphic for infinite group,
778;
number of irreducible accidental singu-
larities of, 778 ;
number of irreducible zeros of, 782 ;
parametric integer for, 784;
quotient of two with same parametric
integer is automorphie, 785.
Third kind, of functions on a Eiemann's sur-
face, 498;
normal elementary function of, 511 ;
moduli of periodicity of, 512 ;
elementary functions of, 511 ;
interchange of argument and para-
meter in, 513;
used to construct Appell's factorial
functions, 533 et seq.
Third kind, of integrals on a Eiemann's surface,
450;
sum of logarithmic periods of, is zero,
451;
must have two logarithmic infinities at
least, 452;
elementary integrals of, 452 (see third
kind of functions);
sum of, expressed by Abel's Theorem, 597.
Third kind of pseudo-periodic function, 321 ;
canonical form of characteristic equa-
tions, 322 ;
relation between number of irreducible
zeros and number of irreducible infini-
ties, 333;
relation between sum of irreducible zeros
and sum of irreducible infinities, 334 ;
expression in terms of cr-function, 335 ;
854
INDEX
of positive class, 335 ;
expressed in terms of simple ele-
ments, 337;
of negative class, 338 ;
expressed in terms of Appell's ele-
ment, 340;
expansion in trigonometrical series,
340.
Thomas, 580.
Thomson (Lord Kelvin), 458.
Thomson, Sir J. J., 649.
Three principal classes of functions on a
Biemann's surface, 498 (see first kind, second
kind, third kind, of functions).
Tractrix and surface of constant negative
curvature, 612, 672.
Transcendental function, 84;
it has 2^00 for an essential singu-
larity, 90;
with unlimited number of zeros over the
whole plane, in form of a product,
92 et seq. ;
most general form of, 99 ;
having simply-infinite system of zeros,
102;
having doubly-infinite system of zeros,
104;
Weierstrass's product form of, 107 ;
cannot have triply-infinite arithmetical
series of zeros, 108 ;
class of, determined by zeros, 109 ;
simple, of given class, 112.
Transcendental integrals, Abel's expression for
sum of, 585 : examples of, 587-590 : of first
kind, 590 : of second kind, 594 : of third
kind, 597.
Transcendents, Abel's Theorems relating to,
579-601.
Transformation, birational, 537-579; effect of,
on irreducible equation, 539.
Transformation, homographic, 625 (see linear
substitution).
Transformation, rational, 537, 579; effect of,
on irreducible equation, 540.
Transformation, uniform, 415 ; birational (see
birational transformation).
Triangle, rectilinear, represented on a half-
plane, 671 ; with special cases, 672 ;
separate cases in which representation is
complete and uniform, 672;
curvilinear, represented on a half-
plane, 685 (see curvilinear tri-
angle).
Trigonometrical series, expansion of tertiary
periodic functions in, 340.
Triply-infinite arithmetical system of zeros can-
not be possessed by transcendental function,
108.
Triply-periodic uniform functions of a single
variable do not exist, 243 ;
example of this proposition, 435.
Two equations of genus, 562-565.
Two-sheeted surface, special form of branch-
lines for, 390.
Two variables, Weierstrass's theorem on
regular functions of, 204-6.
Unconditional convergence of series, 21 ; of
products, 91.
Unicursal equations or curves, 548.
Unifacial surfaces, 372, 380.
Uniform convergence of series, 21 ; of pro-
ducts, 91.
Uniform function, defined, 16.
Uniform function, must assume any value at
an essential singularity, 61, 64, 115 ;
has a unique set of elements in continua-
tion, 68;
is constant everywhere in its region if
constant over a line or area, 72 ;
number of zeros of, in an area, 77 ;
must assume any assigned value, 78 ;
must have at least one singularity, 78 ;
is polynomial if only singularity be
accidental and at infinity, 83 ;
is rational and meromorphic if there be
no essential singularity and a finite
number of accidental singularities,
85;
transcendental (see transcendental func-
tion) ;
Hermite's sections for integrals of, 220 ;
of one variable, that are periodic, 238 et
seq.;
of several variables that are. periodic,
247;
simply-periodic (see simply-periodic uni-
form functions) ;
doubly-periodic (see doubly-periodic uni-
form functions).
Uniform function of position on a Eiemann's
surface, multiform function becomes, 383,
390;
most general, 418;
algebraic equation determining, 419 ;
has as many zeros as infinities, 420.
Uniform integrals of differential equations of
the first order, and their characters : con-
ditions for, 283.
Uniformity of elliptic functions, 233-235.
INDEX
855
Uniformly converging function-series can be
differentiated, 156.
Unity, equations of genus, 554-562.
Unlimited number of essential singularities,
functions possessing. Chap. vii. ;
distributed over the plane, 134 ;
over a finite circle, 140.
Velocity, and velocity potential, in hydro-
dynamics, 639 et seq.
Vivanti, 113.
Von der Miihll, 611, 707.
Von Mangoldt, 753, 788.
Voss, 606.
Watson, 103.
Weber, 223, 625, 753, 767, 771.
Weierstrass, v, vi, vii, 15, 51, 61 et seq., 64, 67,
68, 92 et seq., 118 et seq., 134 et seq., 166,
173, 204, 277, 299, 344, 358, 518, 519,
528.
Weierstrass's il/-test for uniform convergence,
292.
Weierstrass's ^-function, 296;
is doubly-periodic, 297 ;
is of the second order and the first class,
298;
its differential equation, 299 ;
its addition-theorem, 307 ;
derivatives with regard to the invariants
and the periods, 311.
Weierstrass's (7-f unction, 293;
its pseudo-periodicity, 305 ;
periodic functions expressed in terms of,
306;
its quasi-addition-theorem, 307 ;
differential equation satisfied by, 312 ;
used to construct secondary periodic
functions, 328 ;
and tertiary periodic functions, 335.
Weierstrass's f- function, 292;
its pseudo-periodicity, 301;
periodic functions expressed in terms of,
302;
relation between its parameters and
periods, 303;
its quasi-addition-theorem, 307.
Weierstrass's product-form for transcendental
integral function, with infinite number of
zeros over the plane, 92 et seq. ;
with doubly-infinite arithmetic series of
zeros, 107.
Weierstrass's theorem on regular functions of
two variables, 204-6.
Weyr, 103.
Whittaker, 103.
Wiener, 161.
Winding-point, 392.
Winding-surface, defined, 392;
portion of, that contains one winding-
point is simply connected, 395.
Witting, 114.
Zero, equations of genus, 548-554.
Zero of a function on Eiemann's surface, how
estimated in multiplicity, 421.
Zeros of doubly -periodic function, irreducible,
are in number equal to the irreducible infini-
ties and the irreducible level points, 266 ;
and in sum are congruent with their
sum, 267.
Zeros of uniform function are isolated points, 74 ;
form of function in vicinity of, 75 ;
in an area, number of, 75, 77, 82, 86 ;
of transcendental function, when simply-
infinite, 102;
when doubly-infinite, 104;
cannot form triply-infinite arith-
metical series, 108.
Zuhlke, 690.
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