52 MAJOE P. A. MACMAHON: MEMOIR ON THE 



We have to pick out I X's and m /i's and to find the associated factors, 2n-l-m in 

 number, which are linear functions of the quantities a. 



In any such selection of I X's and m /u's there will be i pairs of X's symmetrical 

 about the sinister diagonals and./ pairs of>'s symmetrical about the dexter diagonals, 

 and the associated factors will depend upon the numerical values of i and j. 



Consider then in the first place the number of ways of selecting I X's in such wise 

 that i pairs are symmetrical about the sinister diagonals. 



This number is readily found to be 



n w 



i l-2i 



With these I X's we cannot associate any /u which is either in the same row or in 

 the same column as one of the selected X's. 



Each of the i symmetrical pairs of X's in this way accounts for 2 /A'S, and each of 

 the l2i remaining X's accounts for 2 p's. 



Thus we must select m p's out of 2n-2i-2 (l2i) p-'s, i.e., m /A'S out of 

 2n-2l + 2i fi's. 



We may select these so as to involve / pairs symmetrical about the dexter 



diagonals in 



(n-l+i\/n-l+i-j\ ,j 



\ j !\ m2j I 



This number is obtained by writing in the first formula nl + i,j and m for 



n, i and I respectively, 



and observe that we may do this because the selection of a symmetrical pair of X's or 

 of one of the remaining X's results in the rejection of a pair of /x's which is 

 symmetrical about the dexter diagonals. 



Consequently the 2n2l + 2i possible places for the m /A'S are also symmetrically 

 arranged about the dexter diagonal. Hence the formula is valid. 



We have established at this point that we may pick out I X's involving i 

 symmetrical pairs and m /LI'S involving j symmetrical pairs in 



. 

 \ij\l-2ij \ J J\ m-2j j * 



We must now determine the nature of the 2n I m associated factors, linear 

 functions, of the quantities a. 



In the matrix of the product delete the rows and columns which contain selected 

 X's and /A'S. We thus delete l + m rows and l + m columns. 



Consider the 2nlm remaining rows. There remain in these rows at most 



2n I m 

 elements a, because l + m columns have been deleted, but some of these elements 



