MAJOR P. A. MACMAHOR ON THE THEORY OP PARTITIONS OF NUMBERS. 381 
Section 7. 
Art, 98. It might have been conjectured that the lattice in solido would have 
afforded results of equal interest, but this on investigation does not appear to he the 
case. The simplest of such lattices is that in which the points are the summits of a 
cube and the branches the edges of the cuhe. 
ai a-, ttg a-, a^■, a- is a partition of a number into eight parts, satisfying the 
conditional relations indicated by the symbols > as shown. The descending order is 
in the positive dii’ection parallel to each axis. The H function 
o 
] 
— X, . i- —\o . 1 — ' X., . 1-^ X, 
' ' r/, ■ r/o ■' ru 
-i^^-X-, . 1 - 
fflL X 1 
ff-a 
a.ru 
X, 
is difficult to deal with, and the result which I have obtained too complicated to be 
worth preserving. I therefore put at once 
Xi = X2 = X,. = X,j = X-, = X,; = X; = Xg = .r, 
I divide the calculation into eighteen parts 
Iiesult. 
1 + of‘ p 
( 1 ) (2) (3) (4) (5) (G) (7) (8) 
and seek the sum 
as follows :— 
Conditions. 
ttfi > a; > 
a-, > a.,, ttj > ttj 
a,; ^ a- ^ 
a-i ^ a,, a._, >■ 
+ sr + 
(1)(2)(:!)(4)(5)(G)(7) (8) 
