226 Memoir on the Theory of the Partition of Numbers. 

 such that every solution 



a i> a 2> a 3> a 



is such that 



A I} X 2 , ...... X m being positive integers. 



This arises from the fact that every term 



X^iXj-Xa-* ...... X,* 



of the summation is found to be expressible as a product 



X .......................................... 



(m) n (m) (m) (m) "I Am 



X 



Denoting this product by 



PA,P Ao P A,,, 



1 ] -L 2 ...... r " 1 



the generating function assumes the form 



i-W^ + Q^ + Q^ + .-Q + W^^ 



(i-poa-ps) ...... a-p m ) 



wherein the denominator indicates the ground solutions and the 

 numerator the simple and compound syzygies which unite them. 

 The terms 



Qi (1> > Qi (2) > Qi (3) ...... denote first syzygies 



O (l) O (2) O (3 tsppond 



^52 ' ^2 ' ^2 ..... " " 



Q(l) n <2) O <8) third 



3 S5s ? S5s ...... 55 l111 



The reader will note the striking analogy with the ge/nerating func- 

 tions of the theory of invariants. 



Similar results are obtained as solutions of linear Diophantine equa- 

 tions. 



