MAJOR P. A. MACMAIION ON THE THEORY OE PARTITIONS OP NUAH3BRS. 393 
This is on reduction 
1 - YiY,. 1 - YiYI 1 - X1X2X3.1 - X1XIX3 
a real generating function. 
Art. 114. For s = 4, the conditions are the same, viz. 
2ai > a.! — “1 
and the fl function, where now 
Yi = X1X4, Yo X0X3, 
IS 
n 
b rt ' 
1 
yielding’ the ground fornvs already found by the first method. 
Art. 115. For s = 5, the conditions are 
Oil *^2 — <^3 — “2 5 
2ai > a,, > «!. 
leading to 
where 
and this is 
70 
-1-—Y 1 
1 
nd „ ^ c 
j J 0 * 1 i 
oc ' a 
n 
Yi = XiX„ Y, = X,X„ Y3 = X3, 
1 
^1-—Y 1 
= n 
4- Y.,Y3 .1 - — y 
0 ' if, 
1 
- 1 - abYiY.,Y, .1 - Y.Y ,. 1 - — X, 
h' a " 
= n - 
- 1 - 5Y1Y0Y3. 1 - -F Y2Y3. 1 - &YiY,Y| 
1 - Y^Y^Y^ 
1 - YiY,Y 3 .1 - Y^YoYI. 1 - YiY|Yi. I - YiYlY;! 
_^ x^x.iXijXjXs_ 
1 - XiXoXsXXo • 1 - XXoXfXiXs. 1 - XiXiX'X^X,. l - XjXIX^XfXa 
establishing the ground forms 
XiX2X3X,X5, XiX2XiX4X5 
■y Y2V2V2Y Y Y2Y3Y2Y 
AiA.2A.3Y4A.5, A-iY2A.3A4A5 
3 E 
VOL. CXCII.—A. 
