THEORY OF THE PARTITIONS OF NUMBERS. 



43 



Art. 129. Before proceeding to eliminate g and h, observe that if we now put 

 g = h = 1, we obtain the generating function for the solutions of the first six of the 

 Diophantine equations corresponding to the squares which possess row and column 

 but not diagonal properties. 



Putting g = h = 1 , the generating function reduces to 



X 1 X 5 X 9 X 10 ) ( 1 X 1 X 8 X 8 X 10 ) ( 1 



^io) (1 X 3 X 4 X 8 X 10 ) (1 



(1 

 indicating ground forms 



connected by the ground syzygy 



. X 3 X 4 X 8 X 10 = 



X 1 X 6 X 8 X 10 , 



X 3 X 4 X 8 X 10 , 



corresponding to the fundamental squares 



Ill 







This is the complete syzygetic theory of these particular squares of order 3. 



G 2 



