54 MAJOE P. A. MACMAHON: MEMOIR ON THE 



has the symbolical expression 



or, putting 



sl = o-j, s*-4s + 2 = o- 2 , 



we obtain 



/n\ ni - _ 



for the number of squares such that 



(1) Sum associated with rows and columns is unity ; 



(2) There are I units involving i symmetrical pairs in the dexter diagonal ; 



(3) There are m units involving j symmetrical pairs in the sinister diagonal. 



Giving i and j all possible values we find that the complete coefficient of 



in the product, which we have already ascertained to have the expression 



may be also expressed in the form 



n ~ l H ~ 



fn-l+ 1 

 \ m 



