TIIF, FOr\I>ATIONs OF A NK\V THEORY. 385 



. . . "AIAV-, . . . "Ai^Ft ... "'A.)!." . . , 



~ ~~ A,-/.,, Mi-'/n *\- r i> A,-j>,,Mi-9i. "j-' 



where (pj^jV*, . ! . p 2 <? 2 r, p,<J,r. . . .) is a partition of (pyr . . .), and the 



sum is for alj such partitions and for a particular partition is Cor all ways of operating 

 upon the suffixes with the parts of the partition. Ex. (jr. 



D n /i, //,., = A.,., + A-,, + //,/<,, + /* 10 A 12 

 Taking only tripartite functions for convenience, consider the function 



"A.MI*! h^f, . ''*.,... 



* 



and the operation D Pi , |r , D^,, . . . T> piq , r , ; 



we have D Pi ,, P , D^,,, . . . D M , r , A A-liri k^ . . . h ,..,..= A; 



where (^j^i /W Pi^t) and (X,/^ X^j/j . . X,//,,i/,) 



being partitions of the same tripartite number, 



The operation of D J>i7|r| upon the product splits up as usual into a number of minor 

 operations, one of which, as shown above, is connected with one of its partitions 

 operating in a definite manner upon the suffixes XJ/A^,, X^i'g, X,/M/,. Hence the 

 first row of the lattice has in certain of its s compartments the tripartite parts ot 

 some partition of />]5y'i ; the second row also will have in certain of its compartments 

 the tripartite parts of some partition of p.iW$ ', and finally we must arrive at a lattice 



whose rows are associated with partitions of piWi, PzWz, p/gPi resj)ectively, 



and whose t columns are associated with partitions of Xj/iji^, X 2 /i,j/ 2 , . . . 

 respectively. There is no restriction on the magnitude of the constituents of the 

 various tripartite numbers which appear in the compartments. The lattices thus 

 defined are enumerated by the number A. We may give the lattice a literal form by 

 writing a p b"c T for (por) in a compartment. We then have a theorem which may be 

 stated as follows : 



Monomial products of letters a, 6, c, ... may be placed in the compartments of a 

 lattice of t rows and s columns in such wise that the multiplication of products in 

 successive rows produces ct p '6' 2 'c ri . . . , a" f & 7l c r * ....... ci^bW . . . respectively, 



;md in successive columns produces a^'ft^'o' 1 . . . , a^b 1 ^"' ....... a^'c"' . . . 



respectively in a number of ways enumerated by the number A above defined. 



It is scarcely necessary to observe that 



2p - 2X = Sy - Sp = 2>- - * v = . . . .-= 0, 

 and that only x + t 1 of the s + t literal products are independent. 



VOL. CXCIV. A. 3 D 



