THE FOUNDATIONS OF A NEW THEORY. 377 



viz., A operates upon a power of x by striking out one x, two x'a, three x'a, <fec., in all 

 possible ways and summing the results. Thus 



AX 3 = XXX + XXX + XXJt + XX96 + XXX + XXX + XXX = SiE 2 + Sx + 1. 



This simple fact shows that we may expect a corresponding theory of lattices, and 

 that this is, in fact, the case is seen immediately one introduces the part zero into 

 the partitions of the functions. I have introduced zero parts into partitions in the 

 Memoirs on Symmetric Functions above alluded to, and have imported into the theory 

 the corresponding operators d and D .* It was there shown that, if n be the 

 number of quantities of which the symmetric functions are formed, 



and thence it appears that we have operations D , d , 1 + D corresponding to the 

 operations A, d/dn, E of the calculus of finite differences. 



Considering partitions which only involve zero parts, we have only finite difference 

 operations ; if we have other integers, we have mixed operations drawn both from 

 the finite and infinitesimal calculus. 



The partition (0*) is derived from 



tetjctjaj . . . a. p 9 



/n\ 

 by putting q = 0, and obviously has the value ( ] and, in the paper referred to, it 



has been shown that D operates upon a monomial by erasing one zero part from its 

 partition, so that 



Do(0>) = (O'- 1 ), 



which is to be compared with the operation of A, viz. : 



where cc " 1 = x(x 1) (x 2) . . . (x m + 1) 



in the notation of the finite calculus. 



Further, it has been shown that D operates upon a product of monomials through 

 its partitions 0, 00, 000, 0000, . . . which are infinite in number, viz. : we are to 

 strike out one zero, two zeros, three zeros, &c., in all possible ways ; but in any one 

 such operation not striking out more than one zero from any monomial factor. 



Ex. gr. D (0) (0*) (0) = (O 2 ) (0*) (0) + (0 s ) (0) (0) + (O 3 ) (O 2 ) 



+ (0 2 )(0)(0)+(0*)(0') + (0 8 )(0) 

 + (0 2 )(0) 



the successive lines being due to the partitions 0, 00, 000 respectively. 



* 'American Journal of Mathematics,' vol. 12, second memoir, "On a New Theory of Symmetric 

 Functions," p. 71 cl seq. ; and vol. 13, third memoir, &c., pp. 8 ei wj.. 

 VOL. CXCIV. A. 3 C 



