ALGEBRAIC FUNCTIONS WITH AUTOMORPHIC FUNCTIONS. 
19 
The differential equations (1) and (2) determine t in terms of z in the tivo cases 
respectively. 
The constants ho, . . . are as yet undetermined. The reason is, that we 
have not yet made any use of the condition which in fact does determine them ; 
namely, that all the projective substitutions, w^hich t undergoes when the independent 
variable 2 of the differential equation describes a circuit round one of the singularities, 
are such as to leave unchanged a certain circle. This circle is, in the figure we have 
drawn, the real axis of the variable t, which is unchanged by all the substitutions of 
the group ; but it may more generally be any circle in the ^-plane. This condition 
will be shown in § 7 to be equivalent to the determination of [n — 1) complex 
quantities, which are the constants h^, ho, . . . h,^_i. But a further consideration of 
this is deferred to § 7 . For the present we shall suppose h^, ho, . . . /c„_i determined in 
such a way as to give the required representation. 
§ 6. Application of the Preceding Theory to the Uniformising of Algebraic Forms. 
We have proved that the genus of groups of the kind we have found is zero, and 
hence the automorphic functions of the group as it stands will not uniformise 
algebraic forms whose genus is greater than zero. But we can find sub-groups of 
the original group, and these will be found to be of genus greater than zero. 
The process of deriving these sub-groups is analogous to the method of building 
up a Riemann surface of any genus by superposing a number of plane sheets and 
connecting them along branch lines. We ioin tos^ether a certain number of tlie 
polygons in the figure, and regard them as forming one new polygon. This will, in 
certain cases, be the fundamental polygon of a sub-group of the original group, and 
may have a genus greater than zero. 
Consider a double polygon, made up by taking together the original polygon, and 
the polygon derived from it by transforming with the substitution S„+i, and erasing 
the boundary which separates them. The new polygon has 2n-sides. By erasing 
all the lines corresponding to the line already erased, we obtain a division of the 
half-plane into 2w-gons. The opposite sides of the 2n-gon are easily seen to be 
transformed into each other by the n substitutions 
_ Q Q _ Q Q _ Q Q 
respectively. 
This 2 n-go 7 i is a ^‘fundamental regioii'ffor the group generated from the substi¬ 
tutions Tj, T2, . . . T„. We proved in § 2 that the group generated by Tj, T2, . . . T„,, 
is a self-conjugate sub-group of the group formed by Si, S2, . . . S,,+2; and that any 
substitution of the latter group is equivalent to a substitution of the former group 
acting on either the identical substitution or on S„+i. This corresponds to the fact 
that a point in any of the derived 2n-gons cam be obtained by transformation with 
D 2 
