THEORY OF THE PARTITIONS OF NUMBERS. 49 



we obtain the corresponding square 



2000 



0011 

 1 1 ' 

 0110 



These examples are sufficient to establish the validity of the theorem. 

 . Art. 138. If we wish to make any restriction in regard to the numbers that appear 

 in the s ih row, we have merely to strike out certain terms from the function 



7, () 



n w . 



E.g., if no number is to exceed t, we have merely to strike out all terms involving 

 exponents which exceed t. 



If the rows are to be drawn from certain specified partitions of w, we have merely 

 to strike out from the functions 



h (1) h <2) h (n) 



"'HI ) n 'w 



all terms whose exponents do not involve these partitions. 



We have thus unlimited scope for particularising and specially defining the squares 

 to be enumerated. 



Let us now consider the enumeration of the fundamental squares of order n, such 

 that the sum of each row, column and diagonal is unity. Observe that if the 

 diagonal properties are not essential the number is obviously n \ 



Art. 139. It is convenient to consider a more general problem and then to deduce 

 what we require at the moment as a particular case. I propose to determine the 

 number of squares of given order which have one unit in each row and in each column, 

 and specified numbers of units in the two diagonals. 



Consider an even order 2n, and form the product 



X (a, + Xa 2 + a 3 + . . . 

 x (a, + a 2 + Xa 3 + . . . 



- 2 + Xa 2)l _i + a 2n ) 

 x (/*! + a 2 + 3 + . . . + a 2;i _ 2 + .,_! + Xa 2 ,,). 

 We require the complete coefficient of 



when the multiplication has been performed. 

 Writing Sa = 



VOL. CCV. A. H 



