20 
MR. B. T. WHITTAKER ON THE CONNEXION OF 
the substitutions T from a point in either the original (n + l-)-gon or the (n + l)-goii 
derived from this by the substitution S,i+i. 
We have, therefore, obtained a new division of the half-plane into 2n-gons, and 
found the group of substitutions corresponding to it. We can now find the genus p 
of this group. 
The opposite sides of the 2?i-gon are transformed into each other by substitutions 
of the group. If we suppose the 2n-gon iifted up from the plane, and opposite sides 
pieced together, we obtain a surface of connectivity (^^ + 1). If n is even, this 
surface is of genus p where n ~ 2 p. In what follows we shall suppose n even. 
Hence, the algebraic relation hetiveen any two automorphic functions of this group 
is, in general, of genus p = ^n. 
The function 2:, which has been obtained, takes every value once in each 
[n + 1 )-gon ; and therefore it takes every value twice in each 27 ?,-gon. But this is 
the condition that the algebraic form, made up of the automorphic functions of the 
group, should be hyperelliptic. 
Hence, the algebraic form, which is made up of the automoiphic functions of the 
group, is hyperelliptic, and of genus ^n ; and, as 2; is a variable which takes every 
value twice in each polygon, the form consists of rational functions of 2; and u, where 
u is a function of 2 defined by an equation 
id = (z - cq) (z — a,) ... (2 — rq+2), 
where tq, a.,, . . . a^+z are constants to be determined. But the function 
\/{2 — edj (2 — e.,) . . . (z — 
is an automorphic function of the group, for it has the same value, save for a change 
of sign, at corresponding points in adjacent {n + l)-gons, and therefore the same 
value at corresponding points in different 2n-gons. 
Hence 
Cii = Cl, a~i 62, . . . Ctn+2 — ^n + 2> 
and we see that the automorphic functions of the group generated from the substitu¬ 
tions Ti, T2, . . . T„ are the algebraic functions of the form defined by the equation 
IT = (2 — Cl) (z — e,) . . . (z — e„+2). 
Thus we have the solution of the problem, “ To fnd a variable of which the 
functions rational on the Riemann surface of the equation 
%d = (z — Cl) (z — e.^ . . . (z — e„+o) 
are uniform functions.” 
