ALGEBRAIC FUNCTTONS WITH AUTOMORPHIC FUNCTIONS. 
II 
tiition S1S2, this second derived polygon adjoins the first along its free side 
through D,,i+ 2- If again we derive a fresh polygon from the original one by 
applying the substitution 818,83, this third derived polygon adjoins the second 
along its free side through Proceeding round 1)^+2 this way, we obtain 
at last a polygon which is derived from the original one by the substitution 
8„,iS,... . 82Si8.,i8,, ... S281. 
But since 
8»+iS„. . . 82S, = S,+2, and S'^+2 = 1, 
this is the identical substitution; in other words, the 2 [n polygon as we go 
round A is the original polygon. 
In the same way we can prove, that at every corner 2 (tz + 1) polygons meet. 
The sides of the polygons are all ^lortions of circles orthogonal to the real axis. As 
we approach the real axis, the polygons become smaller and more crowded together. 
If from the original polygon we derive others, hy transforming it loith all the 
substitutions of the group generated hy 8], 82, . . . 8,1 + 0, loe cover the half-plane once 
and only once. 80 the original polygon is a “ fundamental region ” for the group of 
substitutions 
In the annexed figure, the polygons in a portion of the plane are drawn to scale 
for the group formed from the substitutions 
3 
253 ^ - 2061 \ 
33 ^ - 253 / ’ 
Q _ 132i - 1675\ 28U - 4786\ 
“ V ’ 11^ - 132 / ’ ~ V’ lit - 281 j ’ 
which are self-inverse substitutions satisfying the required relation 
8,8.283848386 = 1. 
Here n, = 4 ; the double points are given by 
D, = 5 + 7 i, D .2 = 2 i, D 3 = 5 4- 2 i, D 4 = 1 ) 5 = 12 -f- x/jji- 
The vertex at the intersection of the 8, and 82 circles is at the point t = 1 i. 
8ince the polygons are conformal representations of each other, they are equi¬ 
angular to each other. 
From the construction of the polygon, all the angular points are equivalent in 
respect of the group. 
The sum of the angles round any vertex is 27 r ; but these angles are the conformal 
representations of the angles of a polygon, taken twice. Hence the sum of the angles 
of any polygon is n. 
