the number of partitions of n 209 



§ 3. Proof of (2). The proof of (2) is very similar. We have 

 (13) a-\(l-x){l-af)il-x^..:f 



= x- (1 - 3a; + 5x' - 7a-« +...)- 

 = S (- 1)'^ + " (2//. + 1) (2i/ + 1) a,-+i'^<'^+i> +^v(. + i)^ 

 the summation now extending from to oo for both jx and v. If 



2 + l/^(^ + l) + iz/(z; + l) = 0(mod. 7), 



then 16 + 4/A(ya + l) + 4z/(z7 + l) = 0(mod. 7), 



(2/A + l)" + (2i^-l-l)- = 0(mod. 7), 



and 2/x + 1 and 21^ + 1 are both divisible by 7. Thus the coefficient 

 of .^■™ in (13) is divisible by 49. 

 Again, all the coefficients in 



(l-a;7) (l-a;») (l-a;^!)... 



[{I - x)J\ -^x^) {V-a?) ... Y 



(except the first) are multiples of 7. Hence (arguing as in § 2) we 

 see that the coefficient of a'"" in 



(l-^-)(l-^')(l-^')--- 

 is a multiple of 7 ; which proves (2). As I have already pointed 

 out, (5) is a corollary. 



§ 4. The proofs of (4) and (6) are more intricate, and in order 

 to give them I have to consider a much more difficult problem, 

 viz. that of expressing 



p {X) + jj (\ + h) X + j) (A, + 2S) X + . . . 



in terms of Theta-functions, in such a manner as to exhibit ex- 

 plicitly the common factors of the coefficients, if such common 

 factors exist. I shall content myself with sketching the method 

 of proof, reserving any detailed discussion of it for another paper. 

 It can be shown that 



(14) ^^ " ^'^ ^^ " ^"^ (1 - ^1 . ^. 1 



{l-x''){l-x^){l-x'^) ... ^-'-x' -^X' 



_ |-^ - Zx^ + or (^ -' + 2x^-' ) + ^^(2g-^ - ,rp) + x^{3^-' + xl')+5x^ 

 ~ ^ -' - 1 1 X - x'^' ' 



where P = ^^ ^ " ^1 ( ^ Z ^(LtI^L • ' 



^ {\-x^)(l-x'){l-x''){l-af)...' 



the indices of the powers of.'?;, in both numerator and denominator 



15—2 



