118 Mr Mordell, On M^- Ramanujan's 



Let also p be any prime number; then we may take 



(&)i, pa>^, («! + CDa, p(o<^ . . . (ft)i + ( jO — 1) fUo, J9&)y), ( ptWi , &),.) 



as the reduced substitutions of order j)- Then for many modular 

 forms* it is well known that unities ^, fu, ^i, ••■, |>-i can be 

 found so that 



is also a relative invariant of the modular group. 



This is also true of the quotient Q = (fi/fioy^Jwo), which is a 

 modular function of co. Q is really an automorphic function whose 

 fundamental polygon (putting (o = x + Ly)is that part of the upper 

 CO plane bounded by the lines x = ±^ and external to the circle 

 a;2 + 2/2= 1, but we reckon only half the boundary as belonging to 

 the fundamental polygon. The only infinities of Q are given by 

 the zeros of /(&)i, 0^2) = 0, and if these zeros are also zeros of the 

 numerator of at least the same order as of the denominator, 

 it follows that Q has no infinities in the fundamental polygon. 

 Hence Q is a constant, so that (f)~Qf((Oi, &>.,). 



Suppose now that 



where ^1 = 1. Then 



becomes (^)" S's' |.^,r^/^e^-'VP 



and in the examples with which we are concerned all the terms 

 will vanish, because of the summation in X, except those for which 

 s = (mod p), and then the sum will become 



^ pA, 



Hence we have 



Equating coefficients, we find, if s is prime to p, 

 pAsp = Qp^Ag. 



* This fact is intimately connected with the transformation equations in the 

 theory of the modular functions. We may note that it is often more convenient to 

 select the reduced substitutions in different ways. 



