122 Mr Mordell, On Mr Ramanujans 



Hence /.(jt>) = 2 i(- 1)^| 



extended to the solutions of p= ^ + S^rj + Srj- for which 



^ = 2 (mod 3), 17 = 1 (mod 2). 



This * can be written as /2 ( p) = ^v, where p = Su^ + v'-, u is positive 

 and V = 1 (mod 3). Also f\ {p)=0 if p = - I (mod 3). This is 

 Mr Ramanujans result (127). 



If a = 3 we have, from Klein-Fricke, vol. 2, page 377, 



r [(1 - r^) (1 - r«) ...]« = - 12 (p - 7)') rf=+''^ 

 where ^ takes all even values and rj all odd values. Hence 



if p = ^^ + 7)"^, I is even, 97 is odd, and both f and ■?; are positive. 

 Also /3( j9)= if ^ = 3 (mod 4). This is Mr Ramanujans result 

 (123). 



If a = 4, then by Klein-Fricke, vol. 2, page 373, 



r \(l — r^) (1 — r^) . . .? = +2p r^""*"^^*''+^''', 



where ^, rj take all values for which |^ = 2 (mod 3). 

 Hence /4 (^) = ^Sp extended to all the solutions of 



^ = p + 3^77 + '67j\ 



where | = 2 (mod 3). Thisf can be written as /4 ( jd) = 2 (v" — 9vu-), 

 where p = 3w'^ + ^^ tt is positive, and y = 1 (mod 3). This is 

 Mr Ramanujans result (128). 



When a = 6,/6 {n) is known by means of the representations of 

 n as a sum of four squares. Mr Ramanujan has overlooked the 

 fact that in his result (159) 2c^ is —J\{p)' The theorem 



/b (wO/e in) =/6 (mn), 



is due to Dr Glaisher. 



When a = 12, we have Mr Ramanujan's results given as 

 equations (1) and (2) in this paper. 



He also gives results when a = i , |. 



* When ^ is even put ^ = 2v, 7] = u-v, and when | is odd put ^ = du-v,'r) = v-7i. 

 Both these cases are admissible, and we find that p=v^ + Bu" and v = l (mod 3). 

 Also S {-1)^ ^=2v + 2v - {Su -v) - ( -3u-v) = Gv, where now w is taken as positive. 



t See the last footnote. In addition to the two cases there considered, 7/ even 

 is admissible. Put then 7] = 2u, ^— -v -3ii, from which p = v^ + 3ii'^ and v = l 

 (mod 3). 



