MAJOR P. A. MACMAHON ON THE THEORY OF PARTITIONS OP NUMBERS. 395 
Art. 118. For 5= 7, the independent conditions are: 
2ai 2 ol ^ > 2a2 + «4 
2a2 
> 
“4 
a.. 
> 
"3 
“4 
> 
“3 
2ai 
“2 
O-i 
> 
tto 
ay 
> 
ai, 
and these lead to 
Y^.l- 
h-cg 
■ 
S'y,.! 
ccl 
A 
ah 
and eliminating d, f, g in succession 
= n 
1 
a-ce- 
= 1 _ AAl- Yj. 1 - ^ Y.,. 1 - YAh. 1 - -V Yi 
fj ^ d-cj- - he ^ ^ c(b ^ 
9 9 7 1 5 
> 1 _ ^ Y 1 _ Y.,YAh • 1 - ^ Y..Yi. 1 - 4 Yi 
g ^ ac - ‘ he ^ ah 
= a 
] 
== 1 - ahccY,Y.Y,Y ,. 1 - ^ Y., Y,Y,. 1 - j- Y .^,. 1 - 4 Y, 
ac be ah 
and eliminating e 
= n 
1 - «kYiY2Y3Yi • 1 - Z/'cYAlYAi • 1 - 4 Y,Yi • 1 
and eliminating c 
o 
_1 - ft^6-YAiYAl_ 
- ahY,Y.Y.Y ^. 1 - a-Y,YA1Yl • 1 - 6'Y,YiY;iYi ’ 
-770' 
And on further reduction it is finally 
1 - YAHYA^I ■ 1 - YAlYfY^ 
- Y AsYAz • 1 - Y,Y2YA1 • 1 ~ YAaYlYi ’ 
- Y.YiYA! • 1 - YAAlYi. 1 - YAiYAl 
establishing the ground forms 
