190 
MATHEMATICS: G. H. HARDY 
a divisor of 8q a even, and 5i an odd divisor. When 5 exceeds 8 the formulae 
are less simple, and involve arithmetical functions of a more recondite 
nature. Liouville gave formulae concerning the cases 5 = 10 and 5 = 12, 
and Glaisher 1 has worked out systematically all cases up to s = 18. More 
recently important papers on the subject, to which I shall refer later, have 
been published by Ramanujan 2 and Mordell. 3 In the latter paper the whole 
subject is exhibited as a corollary of the general theory of modular in- 
variants. 
The primary object of my own researches has been to deduce the formulae 
for s = 5 and s = 7 from the theory of elliptic functions, and so to place the 
cases in which 5 is odd and even, so far as may be, on the same footing. The 
methods which I use have further important applications, but this is the one 
which I wish to emphasize at the moment. The formulae which I take as my 
goal are the formulae 
given by Bachmann (pp. 621, 655). Here n as an odd number not divisible 
by any square (so that there is no distinction between primitive and imprimi- 
tive representations); m runs through all odd numbers prime to n; B is 80, 
160, 112, or 160, according as n is congruent to 1, 3, 5 or 7 (mod. 8); and C 
is 448, 560, 448 or 592 in similar circumstances. These formulae are the cen- 
tral formulae of the theory: they involve infinite series, but these series are 
readily summed in finite terms by the methods of Dirichlet and Cauchy. 
With them should be associated the formula 
where A is 24, 16, 24, or 0: but this formula, as we shall see, stands in some 
ways on a different footing. 
2. My new proof of the formulae (1) and (2) was arrived at incidentally in the 
course of researches undertaken with a different end, that of finding asymptotic 
formulae (valid for all values of s) for r s (n) and other arithmetical functions 
which present themselves as coefficients in the expansions of elliptic modu- 
lar functions. In a paper 4 shortly to appear in the Proceedings of the London 
Mathematical Society, Mr. Ramanujan and I have developed an exceedingly 
powerful method for the solution of problems of this character, and applied 
it to the study of p(n), the aumber of (unrestricted) partitions of n. This 
method, when applied to our present problem, introduces the function 
a) 
(2) 
