46 MAJOR P. A. MACMAHON: MEMOIR ON THE 



2 1 

 2 1 



1 2 



connected by the fundamental syzygies 



120 102 111 201 021 

 012 + 210=2111= 012 + 210. 

 201 021 111 120 102 



Art. 133. If the sum of each row, column, and diagonal be 3, the number of 

 general magic squares of order 3 that can be constructed is, from the generating 

 function, the coefficient of ar*" in the expansion of 



(l-.x- H )-(l-;r<)-'\ 

 and this is found to be 



n 2 +(n+l) 2 . 



Art. 134. The ordinary magic squares, the component integers being 0, 1, 2, 3, 4, 

 5, G, 7, 8, are eight in number and are easily found to be 



723 

 =048, 

 561 



and seven others obtained from 



120 201 120 021 120 021 



3012 + 012, 012+3210, 3012 + 210, 



201 120 201 102 201 102 



102 201 102 201 102 021 

 210+3012, 3210 + 012, 210+3210, 

 021 120 021 120 021 102 



021 021 

 3210 + 210. 

 102 102 



Art. 135. There is no theoretical difficulty in proceeding to investigate the squares 



