394 MAJOR P. A. MAC:\[AHON OX THE THEORY OF PARTITIONS OF NUMBERS. 
connected by the simple syzygy 
(X.X.XaX^X,) (X,X?X^X1X,) - (X,X,X3%X,) XiXiXlX^X,) = 0. 
Art. 116. I stop to remark that one of these ground forms, viz.:— 
x.x^xix.x, 
is new, not having so far presented itself in a partition theorem. It is one of an 
infinite system which merits, and will receive, separate consideration later on. The 
one before us is associated with partitions at the points of the disloca,ted lattice. 
A ^ ^ A 
,L 
r'^ 
Art. 117 For 5=6, the conditions are : 
leading to 
where 
This is 
2a.j ^ -p “a 
2ai > a.y 
“a — 
n-^- 
- 1 - Yi.l - ^ Yb.l 
a be 
Y., 
Y, = X,Xa, Y, = X,Xa, Y3 = X3X4. 
1 
n- p -^— 
- 1 - — Yi. 1 - 4^ YAh. 1 - - Y 
a O' a 
1 
= n-^- 
- 1 - Ys. 1 - -4- YAb. 1 - Z^YAbY. 
1 - Y1Y2Y3.. 1 - yaIY' . 1 - yaiy 
establishing the ground forms : 
XiX3X3X,X3Xo 
X V2V2 V2V2V 
l-A. 2 -A. 3 -A. 4 -A. 5 -A 6 
-A. 1 .A. 2-A. 3 A. 4 A. 5 wA g 5 
unconnected by any syzygy. 
