380 MAJOR P. A. MAcMAHON ON COMBINATORIAL ANALYSIS. 



the famous Problem of the Latin Square. I anticipate what follows to the extent 

 of observing that the Latin Square again presents itself without special effort on 

 the part of the investigator, and that a new and very simple solution of that and 

 associated problems is obtained. 



I seek to obtain theorems which flow from a consideration of symmetric functions 

 of several systems of quantities, taken in conjunction with appropriate operations. 



I make the reference MAcMAHON, " Memoir on the Boots of Systems of Equa- 

 tions," ' Phil. Trans.,' A, 1890. 



Consider the systems of quantities 



a l> a 2> a 3> ' 



and write 



Denote the symmetric function 



by 



so that a pv . . . . = (100 . . .' 010 . . .' 001 . . / . . .) 



Ex. gr. a mi =- Sa^ysS* = (1000 0100 0010 0001) 



The quantities a pqr . . . are the elementary symmetric functions. 

 The linear operator d nicp ... is defined by 



JP ,,r, . . d 



p ... 2 ,.-,,,_. r-p, . . . J., . 



so that c?,oo = 



'100 . . . 



^OOl . . . = S 



1 

 and then D w , r 



plqlr 



,>... *,,... 



the multiplication of operators being symbolic as in TAYLOR'S theorem, so that 

 Dp ?r . is an operator of the order p + q + r + . . . and does not denote 



