MAJOR P. A. MACMAHON ON THE THEORY OF PARTITIONS OF NUMBERS. 387 
(^s "i” —i —l) ~1~ — 
“1~ {('s—n ' 71 —l) ^3—m—l) ij^s — l~n ~~ ^s—l — n—l) 
on the dexter, and again the two powers are equal. 
Hence the identity under consideration is established, and this carries with it the 
proof of the diagrammatic representation of the generating function on the points of 
the solid reticulation. 
Art. 102. I resume the general theory of the partitions on the summits of a cube. 
When the parts are unrestricted in magnitude the generating function has been 
found. A process similar to that employed leads to the theorem that when the parts 
are restricted not to exceed t in magnitude the generating function is the quotient of 
I + a (2.x“ + + ox^ + + 2a;®) 
+ a? (x® 4- 3x® + 4x^ + Sx® + 4x® + 3x*® x^^) 
-f- a® (2x^° + 2x“ + + 2x^® 4- 2x’^) 
+ . x^® 
by 
(1 — a) (1 — ax) (I =• ax®) (1 — ax®) (1 — ax^) (1 — ax®) (1 — ax®) (1 — ax’’) (i — ax®), 
the required number being given by the coefficient of a*x“. Denoting the numerator 
by 1 + aP (x) + a®Q (x) + a®R (x) a* . the whole coefficient of a' is 
(9)(10)...(< + 8) , , (9)(10)...(f+7) , (9) (10) +6) 
( 1 )( 2)...(0 + ' Hi )( 2 )...((- 1 ) Hi)(2)...(7-2) 
+ Pt (x) 
(9) (10) ■ ■ ■ (^ + 5) 
(1) (2) ... p - 3) 
+ x'®. 
(9) (10) . . . (t + 4) 
(1) (2) 
Denoting this generating function by^F^ (x), I find 
P(x) = 
F, (x) 
0 
( 1 )’ 
Q (x) = 
K(x) = 
Xi5 = 
F. (x) - F, (x) + X 
Fa (x)-^^F,(x) + x 
F4 (0-(^F3 (x) + x 
(8)(9) 
(1)(2) ’ 
(8) (9) 
( 1 )( 2 ) 
Fi (x) — X® 
(8) (9) 
( 1 )( 2 ) 
F 2 (x) — X® 
(7)(8)(9)_ 
(1)(2)(3) 
(7) (8) (9) 
(1)(2)(3) 
F, (x) + X® 
(8) (7) (8) (9) 
(1)(2)(3)(4)’ 
whence 
F.(x) = - 
(6)(7)(8)(9) 
(1)(2)(3)(4) 
Fi (x) + x^° • 
(5) (6) (7) (8) (9) 
(1)(2)(3)(4)(5)’ 
3 D 2 
