192 
MATHEMATICS: G. H. HARDY 
We then find 
(9) 
the summation now extending to k = 0, 1, 2, . . . and all h of opposite parity 
to k. This is our fundamental formula, when s = 5. Two steps remain: 
first, to prove the identity of 05(g) and # 5 ; secondly, to deduce the formulae of 
Smith and Minkowski. 
4. The first step presents no very serious difficulty, for it involves nothing 
beyond an adaptation of the ideas used by Mordell in his paper quoted in §1. 
We prove first that 9s behaves like # 5 in respect to the linear modu- 
lar transformations r = r + 2, r = — 1/T; so that 9 5 /# 5 is an invariant of 
the modular sub-group called by Klein-Fricke and Mordell r 3 . Secondly, by 
studying the transformation r = (T — 1)/T, we prove that 6 5 /# 5 is bounded 
in the 'fundamental polygon' associated with T 3 . It then follows that the quo- 
tient is a constant which is easily seen to be unity. In all this the only 
difficulty arises from the use of certain reciprocity-formulae satisfied by Gauss's 
sums. 
We now transform (9) by effecting the summations with respect to h, 
using certain contour integrals of a type common in the work of Lindelof 
and other writers. We thus obtain 
a fundamental identity which contains the whole theory of the representation 
of numbers by sums of 5 squares. The symbols j and n alone require expla- 
nation; j runs through the complete set of least positive residues of 0, l 2 , 
2 2 , . . . ,(k — l) 2 to modulus k, each taken as often as it occurs; and i±k 
is the multiple of k deducted in order to arrive at such a residue. And the 
remainder of the work is purely arithmetical. Picking out the coefficient 
of q n , we obtain a series for r&(n) which is found, after some reduction, to be 
equivalent to the series given by Bachmann. 
4. The formulae which correspond to (10) for s = 7 and s = 3 are 
3 ( 1,3,5,... j w = 0 
mk +j 
(10) 
2,4,6,... j m = 0 
