386 MAJOR P. A. MACMAHON ON THE THEORY OF PARTITIONS OF NUMBERS. 
I say that this generating function may be exhibited by factors placed at the 
points of the lattice in solido. These factors are of form 
(s_+ 1) 
’ 
and such a factor must be placed at every point which is the s'” occurring along a 
line of route in the cubic reticulation. 
I take I, m, n in ascending order, and remark that the number of points possessing 
this property is the coefficient of x" in the product 
X [I X x'-^^) [l X x~ . . . + (I + a: + x' + a;"”’), 
which is 
(1 Z ^)3 (1 “ (1 - (’ - ^")> 
and that, if Cj denote the s'” of the third order of figurate numbers, this coefficient is 
^s—l ^s—m ^t—n ~k { —m ~k ^s—l — n “k m—n) 
the term — being omitted, because s is at most ? + w + n — 2. 
I propose, therefore, to prove the identity 
^s—1 ^'s—m ^'s—n “t 1—m ~t "t m—» 
s = j ^ s) 
s=l 
(S) 
^8 ^s—m ^8—n 
s=i + m+n—i 
= n 
S=1 
(g +1) 
(s) 
The factor {I + s) occurs to the power 
^l + s ~k ^s—m ^s—n “k + “k + s — i 
on the sinister side, and to the power 
(^;+s ^l+s-l) “k {^s l) (^s—m ^s—m —l) 
(^s—« ““ ^3—n —l) ~k (^? + s-?n ^i+s—m —l) “k {,^1+e—n ~~~ ^i + i—n —l) 
on the dexter. But 
Cf. Ci._i = h). — k. 
Hence, under all circumstances, the two powers mast be equal. 
Again the factor (.s) occurs to the power, 
- + i>s-l + ^ 5 -m + i>i-n — 
on the sinister side, and to the power 
