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MR. E. T. WHITTAKER ON THE CONNEXION OF 
§ 4, The Automorphic Functions of the Group. 
i^’i'om the fact which has jnst been proved, that the genus of all groups of the 
kind v/e have found is zero, we know that the algebraic relation between any two 
automorphic functions of the group is of genus zero ; therefore all the automorphic 
functions of the group can he expressed as rational functions of a certain one of them. 
We shall denote this one by 2. 
First, let us see what degree of arbitrariness there is in the choice of the function z. 
If a, b, c, d, are any four constants (which can without loss of generality be taken 
to satisfy the relation ad — he = 1), then 
az + b 
cz + d 
is another such, function as 2. Hence the function 2 contains three distinct arbitrary 
constants. 
2 takes every value once, and only once, in each polygon of the figure. The three 
arbitrary constants may be taken to be the place of its zero, the place of its infinity, 
and a multiplicative constant. 
Now consider the conformal representation of a t-polygon on the z-plane. 
The function 2 takes every value once in the polygon ; therefore the conformal 
representation of the polygon will cover the whole 2-plane. Also, 2 takes the same 
value, say at each of the corners of the polygon ; suppose that 2 takes the 
values Ci, c..) 63, . . . e,i+i, at the points D,, Dj, D3, . . . D,j+i, respectively. 
As t describes the boundary of the polygon, beginning at D„+2, 2: begins with the 
value c„+2 and varies until, at Di, the value Cj is reached ; then, retracing the same 
series of values, 2 returns to the value 6^+2 at C]. Then at D2 the value eo is reached, 
and at C2 2 takes the value 6,^+0 again ; and so on round the polygon. 
Thus the conformal representation of the boundary of the polygon is a series 
of lines (not necessarily straight), radiating from the point e-,+2 to the points 
03,. . „ in succession. The polygon corresponds to the whole z-plane, ivith 
this regarded as boundary. tSmall arbitrary variations in the form of the lines 
radiating from e.^^o to c,, e.,, . . . e„+i, merely correspond to small arbitrary variations 
in the boundary of the polygon. 
Thus we see the nature of the solution of the problem: To conformally represent the 
whole plane of a, variable 2, bounded by a set of finite lines radiating from a point, on 
a curvilinear polygon in the plane of a variable t; this polygon being the fundamental 
region of an infinite discontinuous group of real projective substitutions of the 
variable t, and 2 being an automorphic function of the group. 
We may note that dz/dt is zero at each of the double points. For if t and t' are 
two points very near a double point, which are transformed into each other by the 
substitution corresponding to the point, we have approximately 
dt' = — dt. 
