THEORY OF THE PARTITIONS OF NUMBERS. 59 



enabling the verification of the results 



o- 4 = cr :j = o- 2 = rr' 2 = <r, = 0, 

 o-/ = 4, <r s o-, = 0, o-V = 2. 



Hence for the even order 2 the whole coefficient is X 2 /n a , corresponding to the only 

 possible square 



and I find for the uneven order 3 



Art. 144. To find in general the number of squares which have two units iu each 

 diagonal we find the coefficient of X 2 ja 2 and obtain for even order 2n. 



<r 4 - 4 96er, 4 ; 



putting n = 2 we find for the order 4 



2o-4 + 2cr/ + 4cr'/+l 60-30-1 



and the verification of this number is easy. 



For the uneven order 2n+l we obtain the number 



'tr/^ 2<r> 1+ (gJcrr^ 



^ (<r 3 a o- 3 + 2o- 3 V 8 ) + ' 



- 



The general value of 



may be obtained by means of the calculus of finite differences. 



There is no theoretical difficulty in finding symbolical expressions for the enumera- 

 tion of general magic squares associated with higher numbers, but the method does 

 not lead to the determination of general magic squares. These must be regarded as 

 arising from the generating function method of 9. 



I 2 



