THE FOUNDATIONS OF A NEW THEORY. 381 



/> + </+>' + successive linear operations. The operation of D p1r upon a 

 monomial symmetric function has been explained (loc. cit.). It has the effect of 

 obliterating a part pqr . . . from the partition of the function when such a part 

 is present, and an annihilating effect is every other case. The operation upon a 

 product has the effect of erasing a partition of pqr . . . from the product, one 

 part from each factor in all possible ways, the result of the operation being a sum of 

 products, one product arising from each erasure of a partition. 



Ex. gr. D^(43 22) = (22). 



If we have to operate with D^ upon 



(32 22) (21 11) 



we have to erase the two partitions (32 11), (22 21), and arrive at 



22) (21 U) = (22) (21) + (32 fl). 



Art. 13. It will suffice to consider three systems of quantities as typical of the 

 general case. 



Take the function 



= (Too A 'oTo'"ooi")(ioo Al oio' H 6or i ). . . 



and the operation D Mfl V M , t . . . D Mf , 



and (ptf!?-! ## 2 r 2 . . . p t g t r,) 

 being each partitions of the same tripartite number. 



. . . p t q,r t ) 



and we have to determine the nature of the lattices enumerated by the number 



A. The tripartite number ( Piq\r^ has a partition of p l -f q l + ^ parts, viz. : 

 100'"' Oio" "OOl" 1 so *^ a *' * n P era *" 1 g w ith D Pl7iri upon the operand, we have to select 

 this partition from the product, one part from each factor, in all possible ways ; the 

 operation breaks up into minor operations as usual, and the first row of the lattice 

 of s columns and t rows will contain in p l + q l + r x of its compartments the tri- 

 partite numbers 100, 010, 001 (p l of the first, q l of the second, and r l of the third) 

 in some order ; the assemblage of numbers in this row is the partition of the 

 elementary function a mri . Similarly a minor operation of D Prtiri produces a second 

 row containing tripartite numbers, the assemblage of which constitutes the elementary 

 function a pt , iV We finally arrive at a lattice such that the tripartites in the 

 successive rows constitute the elementary functions o M|ri , <^ Mfl t respec- 



