THEORY OF THE PARTITIONS OF NUMBERS, 

 which must clearly have the form 



a, a, 



! 2 1 



and we may associate with it the Diophantine equation 



and regard a,, 2 and a 5 as the unknowns. 



The syzygetic theory is obtained by forming the sum 



for 



all solutions of the equation, and the result is 



i 



l_ 



a' 



auxiliary quantity and the meaning of the prefixed symbol fi is that 

 n of the algebraic fraction in ascending powers of X,, X a , X 5 we are to 

 rms only which are free from a. 



Where (6 IS an ctLi^vmc^i y vjucmuiuy CHAH LUC meet 



after expansion of the algebraic fraction in as 

 retain those terms only which are free from a. 

 The expression clearly has the value 



1 



i-x^.i-x-X 



denominator factors denote the ground solutions 



The 



and the absence of numerator terms shows that there are no syzygies. 

 Thus the fundamental squares are 



1 

 1 



1 



1 



and this is otherwise evident. The case is trivial and is introduced only for the 

 orderly presentation of the subject. 



Art. 128. Passing on to the general magic squares of order 3 we have the square 











