MAJOR P. A. MACMAHON ON THE THEORY OF PARTITIONS OF NUMBERS. 353 
such that every solution 
is such that 
“u “29 “39 • • • “9 
+ Xjaf ... 4- ^ 
0.0 = + XzOif'* . . . + 
^3 = XiO.'i^ 4- Xjaf . . . + 
“9 = Xjaf) + Xoa® ... 4 X,^af 
Xj, X 2 , . . . X„^ being positive integers. 
This arises from the fact that every term 
X?‘X?X,?^. . . Xr 
of the summation is found to be expressible as a product 
... xfT' 
X {Xf^Xf^Xf ^. . . Xf 
X. 
X ixf'^xf'^^xf'^ ... 
Denoting this product by 
^pA.i'DAj T>A.„ 
-T 1 JT 2 . . . X 
the generating function assumes the form 
1 - (QS^’ + Qf + Qf + ...) + (Q^^> + QF + Qf +...)- (Qi'^ + ...) + 
wherein the denominator indicates the ground solutions and the numerator the simple 
and compound syzygies which unite them. 
The terms 
Qf\ Q?' • • • denote first syzygies 
Q2^ Q® ... 9, second „ 
Qf, Qf’... „ third „ 
The reader will note the striking analogy with the generating functions of the 
theory of invariants. 
2 z 
VOL. CXCII.—A. 
