9( ; MAJOR P. A. MAcMAHON: MEMOIR ON THE 



Whence 



7r, + 7r.,+7rj = n(n-l) m(m-l) ; 



and 



T / no m n\ r 'Xn-l)'('-l) 



Hl-l)<-l) ( m n J * 

 If, in the succession 



,[ 3|j|li , i|am-l , 



we fix upon any dividing line and arrange the letters to the right of it in alphabetical 

 order, thus obliterating the lines to the right of the one fixed upon, we obtain a 

 permutation involving (suppose) s lines which yields x to the lowest power that 

 occurs in the sub-lattice function of order s. When the lattice is complete we may, 

 in any derived permutation, write m -, +1 for a., and invert the order, and we thus 

 obtain another permutation belonging to the same sub-lattice function as the former. 

 For a succession a. p \ a. q p > </ in the former becomes by the stated operations 

 - f +i|*-,+if where mq+l > m p+1 in the latter; and if a p is the k th letter 

 from the left of the former permutation, ,_+! is the mnk th letter from the left of 

 the latter. Hence, if the power of x given by the former permutation be 



that given by the latter is 



Thus, for every term x p in L., there is a corresponding term x mns ~ p . 

 Hence we may say that L, is centrically symmetrical both as regards the powers of 

 * and the coefficients. 



If e be the lowest power of x in L,, determined as above, the highest power of x 

 will be mnse. 

 Ex. gr., 



M oo,3,3) = l, 



L,( oo, 3, 3) = x a +2x 3 +2x 4 +2x*+2x e +x' ! , 



and it will be noted that 



in L I( x* and *-* ; in L,, 3* and x 18 '* ; in L 3 , of and a"-* ; 

 in pairs, whilst the theorem is clearly verified in L. and in L, 

 The result of writing 1 for x in L. is the acquisition of the factor x by 



occur 



exprlion of' \ " "** eX P nents of * tha * ^ in the 



L. ( co, m> n). Consider again the permutation 





