THEORY OF THE PARTITIONS OF NUMBERS. 57 



The product of a 2 (l \ 2 (2 ' is thus, after re-arrangement, effectively equivalent to 

 pf+ (A + /A-2) (a, + 03.) #tf>,- (X + /4-2) (a^ + ag. 8 ) p. 



4)}a 1 a 8( (a, +*,)/,, 



2! 2 . 

 Regarded apart from > 2 , ^ this expression is a function of a,, a 3 , ; the product 



a^a/- 1 ' 

 is a function of 2 , a 2 -i> and generally the product 



is a function of at,, a a ,, +1 _ s , and all of these products are of similar form in regard to 



Pa, Pi, ^, P- 



Remembering that we desire the coefficients of 



(!,... a 2 ,) 2 

 in the product 



we must distinguish between p 2 where it occurs as a multiplier of af' + a^ and where 

 it occurs as a multiplier of a^a,,, and make a similar distinction in respect of ^, 2 . 

 Put then 



(tti' + as,, 2 ) pi 2 = (ai 2 + a ,, 2 ) H^. 

 Putting further the quantities a equal to unity and regarding a product 



PaP\irfir^ 

 as a symbol for the coefficient of symmetric function 



/0+rfli + -'-\ 



in the development of symmetric function 



(i 2 )- + "(ir 2 ', 



I say that 



> 1 -2 (X +j u-2)^ 3 + 2 (X-l) (/*- 



is the symbolic expression of the required coefficient of 



VOL. CCV. - A. 



