THEORY OF THE PARTITIONS OF NUMBERS. 109 



Putting pi = p t = . . . = p n = p for a complete lattice, we find 



PF O =I, ib(?'Hi 



?F .. f(p+l)...(p+s)1 (np+l) J(p-H)...(p + g-i)i 

 (!)...() (1) (!)...(-!). 



A simplification, when n = 2,. is interesting ; for then 



In fact, more generally it will be found that 

 (a vA- ^(p)...(P 



An interesting verification is supplied by a result in a previous paper.* It was 

 therein shown that the number of permutations of the letters composing the product 



which have s dividing lines, is the coefficient of X'a^yS 7 in the expansion of the product 



From this expression I derive a function of x, viz., 



and therein the coefficient of X'a* 1 /? 7 is readily shown to be 



.(pK..( 



{(I)- } a 

 as already obtained. 



When the lattice is complete the functions PF. possess elegant properties, just as 

 when the permutations are restricted. 



* " Memoir on the Theory of the Compositions of Numbers," 'Phil. Trans.,' A, 1893, Art. 24. 



