32 
ON ALGEBRAIC FUNCTIONS AND AUTOMORPHIC FUNCTIONS. 
order that the group of substitutions of t may be discontinuous, i.e., in this case may 
conserve a fundamental circle, is equal to the number of the constants k. 
In order that a self-inverse substitution with complex coefficients, 
at -J- lj\ 
may leave unchanged a given circle, two of the four real constants contained in the 
substitution must be determinate in terms of the others. 
Now there are [n + 2) fundamental self-inverse complex substitutions, containing 
4 {n + 2 ) real constants ; of these, the relation 
S1S2S3 
S 
,1 + 2 
= 1 
accounts for six. So (2n + 1) of the real constants are determined in terms of the 
other (2n -f- 1) by the condition that the group is to conserve a fundamental circle ; 
but as the fundamental circle may be any whatever, and so involves three constants, 
we must deduct three from the number of equations, giving (2n — 2). Thus, 2 n — 2 
real, or n — 1 complex, constants can be determined from the condition that the 
substitutions of t conserve a fundamental circle. This accords ivith the fact, otherwise 
arrived at, that the constants k^, ko, . . . Xy_i, in the differential equation have to he 
determined from this consideration. 
Among the quasi-uniformising variables of any algebraic form there are several 
distinct uniformising variables. The groups we have found in § 3 have simply- 
connected fundamental polygons. But automorphic functions exist, for which the 
fundamental polygons are multiply-connected. 
The simplest example of such a function is 
where P is Weierstrass’ elliptic function with periods and 2m,; the fundamental 
polygon is the space between two circles in the Gplane. 
The automorphic functions studied by Schottky, Weber, and Burnside may be 
regarded as generalisations of this. As these uniformising variables with multiply- 
connected fundamental polygons are included in the general set of quasi-uniformising 
variables, they are defined by the same differential equations as the uniformising 
variables with simply-counected polygons, exce})t that the constants k will have 
different values. 
