50 MAJOR P. A. MACMAHON: MEMOIR ON THE 



the product is, taking the th and 2n+l-t th factors together, 



-!) a 2 +(X-l) a 2B _i 



Observing that we only require terms which involve the quantities a with unit 

 exponents, the product of the first two complementary factors is effectively 



and the complete product 



has, on development, the form 



s 2 " + A,* 2 "- 1 + A 2 .s- 2 "- 2 + . . . + A 2 ,,, 



where A,,, is a linear function of products of the quantities a, each term of which 

 contains m different factors a, each with the exponent unity. 

 Since, moreover, x"' gives rise to the term 



ml ^0.^2... a,,,, 



it follows that the coefficient of Sia 2 ...a a ,, in the product is obtained by putting each 

 quantity a equal to unity and s m = m\. 



Hence, if S" = ml symbolically, the symbolic expression of the coefficient is 



or 



or writing 



-s 2 4.S + 2 = o- 2 , .s 1 = 



This is the complete solution of the problem for an even order 2n. 

 For an uneven order 2+ 1, it is now evident that the symbolical expression of the 

 coefficient of 



is 



{ o- 2 + 2 (X + /i) a, + X 2 + ^Y (o-i + V) 



the complete solution in respect of the uneven order 2n+l. 



