376 MAJOR P. A. MAfMAHON ON COMBINATORIAL ANALYSIS. 



Let any operand 



(V/tiV . . .)(V/*zV -MV/^V ) 

 so multiplied lie denoted by 



fkjV ...)... 



then we have the following law of symmetry : 

 From a given finite assemblage of numbers 



^l> /*! v \-> ' ^2> /*2 V 0' ' ^3' /^S) W 3 



construct all the products 



(V^V . . .)(W' . . -MVMsV .-).-. 



which have a given specification (Xfiv . . .) and all the products 



(PiWi -)(?W2 -MjWs ..-) 

 which have a given specification (pqr . . .). 



If 



2Co(x 1 >,v .)(^> 3 V .MXaVsV ...) = + A 



then 



the lattices being derived from 



This is the most refined law of symmetry that has yet come to light in the algebra 

 of a single system of quantities (cf. " Memoirs on Symmetric Functions," ' Amer. J.,' 

 loc. cit.). The actual representation of the things enumerated by the number A is 

 obtained with ease by this theory of the lattice. 



2- 



Art. 8. So far the operations have been those of the infinitesimal calculus, and 

 the numbers involved in the partitions of the functions have been positive integers 

 excluding zero. If we admit zero as a part in the partitions, we find that we have to 

 do with the operations of the calculus of finite differences. At the commencement 

 of the paper d/dx was shown to be a combinatorial symbol, in that when operating 

 upon a power of x, the said power being positive and integral, it had the effect of 

 summing the results obtained by substituting unity for x in all possible ways in the 

 product of x's. Now the corresponding operator of the calculus of finite differences, 



