16 
MR. E. T. WHITTAKER OH THE COHHEXTOH OF 
constants, is a solution of the problem (/ 3 ) equally with t, we shall expect t to be 
given by a differential equation of which the general integral is [at + h)l{ct + d) ; 
in other words, by a differential equation of the form 
z} = R(2;), 
where E, [z] is some function of 2, aijd 
^ ^ (dzidty ^ (dzjdty 
is a Schwarzian derivative. 
As [t, z} is unaltered by a change of t into {at + h)/{ct + d), R (2) is an auto- 
morphlc function of the group, and therefore R (2) is a rational function of z. 
We have to find R (2). 
Considering the conformal representation, we see that 2 and t are regular functions 
of each other, except near the points 2 = Cj, C2, . . . e.^+o, co. Hence, except at these 
special points, 2} is a regular function of 2, and we shall not get an infinity of 
R (2). As 2 is a uniform automoiqDhic function of t, — is infinite only at 2 = 00. 
Near 2 = 00 (supposing for the present that no one of the quantities Ci, Co, .. . 
is infinite), 2 and t are uniform functions of each other, so 
2 == y—- 7 h -{■ c {t — ty where a is not zero. 
t 
This gives 
Hence at 2 = 00, ^ 2} must be zero to at least the order 
Near 2 = c,., 2 is a uniform function of t, but dz/dt is zero. So near this point, 
z — e, = c {t — tof d {t — tof + . . ., 
where c is not zero, since t has at the point a simple branch-point, considered as 
a function of 2. 
This gives 
2 ~ 16 cH^ - t,y + • • • — 16 (0 - ef + • • • 
Thus the only infinities of the rational function R (2) are at the points Cj, 6-2,. ■ 
and these points are poles of the kind just found. 
Hence 
R(2 ) = -a-"s , 
r=i y 
- 
+ 
h+2 
V 
r=:l 
^)l + 2 j 
