48 MAJOR P. A. MAcMAHON: MEMOIE ON THE 



In fact, if 



(x-a l )(x-a 3 )...(x-OL a ) = x"-p l X n - l +...+(-) a 



and 



w! D H , = 



an operator of order w obtained by raising the linear operator to the power w 

 symbolically as in TAYLOK'S theorem, then the number in question is concisely 

 expressed by the formula 



D'V, 



a particular case of a general formula given by the author (loc. cit.). 

 Art. 137. To introduce the diagonal properties, proceed as follows: 

 Let /;.,,. ( * ) denote what !>,, becomes when Xa s , /Aa,,_., +1 are written for a n _, + i 



respectively, and form the product hj-^ h w (2 \ . . h,^"\ 

 1 say that the coefficient of 



in the development of this product is the number of general magic squares of order n 

 corresponding to the sum w. 



To see how this is take 'n = 4, w = 1, and form a product 



x 

 and observe that, in picking out the terms 



one factor must be taken every time from each row, column and diagonal ot the 

 matrix. 



Similarly, if n = 2, we form the product 



x { X a a/ + X^a 2 a 3 + ^a z 2 + (Xa 2 + /ua 8 ) (a! + a 4 ) + a^ + ajOt 4 + a 4 2 } 

 x X a a 2 +XAaa + l t 2 a 2 +Xa + xa) (a 1 + a 4 ) + a 1 :J + !< + a/} 



In forming the term involving 



xy % VV 



regard the successive products as corresponding to the successive rows of the square, 

 the suffix of the a as denoting the column, and X, fi as corresponding to the 

 diagonals. 



Thus picking out the factors 



XX 2 , /ua 3 a 4 , y* 2 a 4 , a 2 a 3) 



