392 MAJOR P. A. MACMAHON ON THE THEORY OF PARTITIONS OF NUMBERS. 
and the numerator now indicates the existence of first and second syzygies between 
the ground forms. 
We have the first syzygies 
(A) = X.X, . X1X0X3 - Xi . X,X^X3 = 0, 
(Bi) = X.XoXj . XiX3X3X4 - Xi . XjXlXiX^ = 0, 
(Bo) = XjX|X3 . XJX0X3X4 - XjX^ . X.X^XjX^ = 0, 
and the second syzygies 
X3(B.3) - = 0, 
X4X.,X3 (B^) - XiX^X (B,) = 0. 
Art, 110. For 5=5, the generating function is 
? o o 
(t-yCt.yO • 1 
hhh 
Xo . 1 - 
aA 
'X,.l 
oA' i «iM;i ’i- 
7 '^4 • 1-1 N3 
«1«3&3C CiiPiCidi 
and therejs no difficulty in continuing the series. The obtaining, however, of the 
reduced forms soon becomes laborious. 
Art, 111. There is another method of investigation. Guided by the results ob¬ 
tained let us restrict consideration to the forms 
which are such that 
X^X? 
X- 
a,„ = a 
5-h 1 -m 
This is of great importance, because we are thus able, for any given order, to 
generate the functions of that order alone. 
Put X„,Xs 4 .,_„ = Y,„ and seek . . . 
Art. 112. For s = 2, the generating function is simply 
1 _ 1 
1 - Yj ~ 1 - XjX., ‘ 
Art. 113. For 5 = 3, the conditions 
2 ai > a S; ttj 
1 
lead to 
2,_ 
0 a 
the letters a, h determining the first and second conditions respectively 
* The validity of this assumption will be considered later. 
