ALGEBRAIC FUN'CTIONS WITH AUTOMORPHIC FUNCTION'S. 
31 
general be the variable ^ of §§ 3 and 4 . But the variables r so found will solve the 
problem of conformally representing the 2-plane, regarded as bounded by a number 
of finite lines radiating from a point, on a curvilinear polygon in the r-plane, such 
that the sides of the boundary can be transformed into each other in pairs by 
certain projective substitutions. The variable t is one of these variables, characterised 
by the condition that the infinite group generated from these substitutions is a 
discontinuous group. 
We can, in fact, find the functions T in this case. We must have 
T = U 
and R (2) must be such that T is holomorphic. So the only possible poles of R (2) 
are the places where dzjdt is zero, i.e., the places 2 = Cj, e^, . . . e,j+2- At these 
places dzjdt is zero of the first order ; so {clzjdtY' is zero of the second order, and 
R (2) may have a pole of the second order. 
Therefore 
R (2) 
where 
vd (2 - e,) (2 - C2) ... (2 - 
and I (2) is an integral function of 2. At 2 = 00, dzjdt has a pole of the second 
order, and id a pole of the {n 2 )'*' order. So I (2) may have a pole of the 
{n — 2 )“" order. 
Therefore 
I (2) = hiZ" ^ h^z^'' ^ -j- . . . -j- 
and 
rp _ -f -h tin-\ I dz'd 
~~ id [dtj ' 
Thus if T is the quotient of two solutions of the equation 
d^vjdd + Tv = 0 
and t is defined by the equation 
i [t, 2] = R (2), 
then T is defined by the equation 
1 {t, 2} = R (2) -f 
kiZ^ ^ ® -t- . . . k„. 
u- 
Comparing this with the equation of § 5 , we see that the variables t given by it, 
ivhen the constants ho, . . . h^-i, are arbitrary, are the quasi-uniformising variables. 
We can now prove that the number of conditions which have to be satisfied in 
