388 MAJOR P. A. MACMAHON ON THE THEORY OF PARTITIONS OF NUMBERS. 
and in general 
F, (x) 
(9)a0)- - ■ (< + i) _ iS)(9)...(f + -i) 
{t -4) 
( 1 ) 
Fs (x) 
(7)(S)...(^ + 4) (^-4)(^-.3) 
(l)(2)...(i-2) (1)(2) 
,(6)(7)...a + 4) (<-4)(i-3)(i-2)^ , , 
,„ (5)(6)...(^ + 4) (^-4)(^- 3)(^- 2)(<-l) 
(1)(2)...(0 ‘ (1)(2)(.3)(4) 
Art. 103. This appears to be the most symmetrical form in which the generating 
function can be exhibited, and it may be assumed that the like function for the solid 
reticulation in general will be of complicated nature. The argument that has been 
given shows that the theory of the ?i-dimensional lattice (easily realizable in piano), 
the part-magnitude being limited so as not to exceed unity, is co-extensive with the 
whole theory of partitions on the lattice of n — \ dimensions. 
Section 8. 
Art. 104. The enumerating generating functions that are met with at the outset in 
the theory of the partitions of numbers are such as are formed by factors of the forms 
1 — 
1 — x‘ 
written for brevity 
[n + 6 -) 
(s) 
All those which appear in connection with regular graphs 
in two and three dimensions are so expressible, and the mere fact of such expression 
proves beyond question that the numerator of the generating function is exactly 
divisible by the denominator ; in other words, it proves that the function can be put 
into a finite integral form. It is quite natural therefore to seek the general expression 
of functions of this form, which possesses this property of competency to generate 
a finite number of terms. Moreover, it is conceivable that such a determination will 
indicate the paths of future research in these matters : will be in fact a sign-post at 
the cross-ways. This is the reason why I undertook the investigation ; but, as 
frequently happens in similar cases, the problem proves d posteriori to be per se of 
great interest and to involve in itself a notable theorem in partitions. 
Art. 105. I consider the function 
(n + 1 )“^ (71 4 - 2 )"= (71 + 3 )"’. . . (71 + sy 
which I also write 
( 1 )“' ( 2 )'^ ( 3 ^ 
... xr, 
(s)“ 
