THEORY OF THE PARTITIONS OF NUMBERS. 47 



of higher orders, but even in the case of order 4 there is practical difficulty in 

 handling the H generating function. There are 20 fundamental squares, viz. : 



10. 



Art. 136. The direct enumeration of general magic squares of given order and sum 

 of row. 



Let h w denote the sum of all the homogeneous products w together of the 



magnitudes 



a,, .,,..._!, . 



If h w be raised to the power n and developed, the coefficient of 



is the number of squares that can be formed of order n, so that the sum of each row 

 and column is w, but in which there is no diagonal property in evidence."" 



* "Combinatorial Analysis The Foundations of a New Theory," 'Phil. Trans.,' A, vol. 194, 1900, 

 p. 369 et seq. 



