26 
MR. E. T. WHITTAKER ON THE CONNEXION OF 
Let ( t, 
at + h 
ct A cl 
be any one of the substitutions of the group associated with the 
given uniformising variable t. Then a Thetafuchsian function T of order m is such that 
We have said that T is to be holomorphic (except at the singularities of the group). 
Such functions exist ; for instance, if w be an Abelian integral of the first kind 
associated with the curve, then diu/dt is a holomorphic Thetafuchsian function of 
order one, and (dwjdtY is a holomorphic Thetafuchsian function of order two. 
To prove the theorem, let 
r = v,/v2, ■ 
where and Vo are any two solutions of (1). Then Vi and ty have singularities, 
considered as functions of t, only where T has singularities. But in any one of the 
polygons in the ^-plane, T has no singularities. Therefore, and V2 are holomorphic 
functions of t (except at the essential singularities of the group, which for the 
present we do not consider). 
Also, Vi and dv^dt cannot be zero together at any point ; for if they were, by 
equation (l), would be permanently zero. Similarly for Vj. 
Therefore, at all points p within any one of the polygons in the ^-plane, we have 
expansions beginning with 
Vi = cd {t — to) + .. ., 
where c and d are not both zero, and 
V2 — e -{-/{t — to)+ , 
where e and f are not both zero. 
And we may not have d andzero together, as and n, are independent solutions 
of the differential e(]uation. 
So, at all points except the singularities of the group. 
gives either 
or, 
or. 
_ C cl (t — tp) + ■ . . 
e + f {t — to) + ..• 
T = A + B (^ — ^o) + ■ • • 5 
T — A {t — ^o) “b B (^ — ^o)^ + • • • 5 
— f f + B + C(^ — ^o) + -- « 
I 
T 
