396 MAJOR P. A. MACMAHOX ON THE THEORY OP PARTITIONS OF NUMBERS. 
Yi\2Y3^4 
Y,Y2Y3ll = XiX2X3X!X,XeX, 
Y.YA^lYl = XiXoXiXiXrX^X, 
Y.Y^YiYj = XiX^XiXiXrX^X, 
YiY?Y^Y| = X.X^X^X^X^XsX, 
V V-V^V^ — ^ V-Y3V4V3V-V 
X^X 2-1-3-14 - A- 2 -A. 3 Y 4 -A. 5 YQ-A .7 
connected by the simple syzygies 
(Y,Y2Y3Y4) (Y,YiY^Y0 - (Y.Y^YIYl) (Y^YlY^Yj) = 0, 
(Y1Y2Y3Y4) (Y,Y5Y^Y1) - (Y,Y2Y3Yi) (Y.Y^Y^Y!) = 0, 
and, denoting these respectively by A and B, the numerator term + 
indicates the second, or compound, syz^^gy ;— 
(Y,Y2Y3Y,) (Y,YiY^Yt) (A) - (Y.Y^Y^Yi) (Y.YsY^Yi) (B) = 0. 
Art. 119. I remark that the forms 
Y,Y 2 Y 3 YL Y^Y^YlYj 
are new to partition theory. 
Art. 120. For 5 = 8 , the reduced conditions are 
a2 + as ^ «i + «4 
ai -f a. > a. 
a. 
a.. 
leading to 
fi 
,.r 
a 
n 
n 
2aj > a.., 
a.s > ot^ 
_ Y 1 _ Y 
ch - he ■ 
1 
(I 
-YWY,,1- Y,V. . 1 - y, 
1 
1 - h,/\\Y.^^\ • 1 - 77' . 1 - J Ys^'i • ^ - 7Y 
n 
1 
■ 1 - h\\\:,\\;\\ . 1 - a:aaai • 1 - . 1 - hx^rY^i 
- YiYA'Ad - YtYAAI - YiYAA! + YfYAAI + Y^IYJY] 
1 - YiYoYsY, . 1 ^ YAi^Al • 1 - YiYPlYl 
1 - YiVAAi. 1 - YiVAAi • 1 - ViYAAI 
Y1Y^Y5Y“ 
