662 
MAJOR P. A. MACMAHON ON THE 
(1 — Ct x x) 
(1 — a } a 2 x 2 ) ( 1 — 
( 1 — aya.-ftoX^) ( 1 — 
\ 
a x a 2 
a? 
(1 - 
) i- 
We have to expand this fraction in ascending powers of x, reject all terms contain¬ 
ing negative powers of a x , a 2 , . . . cc m , and then obtain the reduced generating function 
by putting a x = a 2 — . . . = a m — L in the portion retained. This has proved a diffi¬ 
cult algebraical problem. 1 am indebted to Professor Forsyth for a beautiful solution 
which he will publish elsewhere."" He establishes that the reduced generating 
function is 
_1_ 
(1 - x) {(1 - x 2 ) (1 - x s )... (1 - O} 2 (1 - x m+1 ) ’ 
a result which agrees with the prediction. His method is that of selective summa¬ 
tion. He forms the general term of the expansion 
a 
Vl \— lli+vl}— Tll 3 ~- 
1 
n 3 + 
^ n 2 +m 3 — ?i 3 + . . . . 
Qj m 3 —n 3 + . . . . 
3 
* X 
m 1 +» 1 +2(m 2 +)! I )+3(m 3 -Hi 2 )+ . . . . 
and performs the enumeration 
S S S S S S ^1!ll+>llH-2(H! i + nj)+3(H! 3 +)l)j+ .. . . 
m x ?i l m 2 n 2 m 3 n 3 
for all values of m l5 n L , to 2 , n 2> to 3 , n 3 , .which make each of the m expressions 
m l — n x -f to 2 — n 2 -f- 
m. 2 — n 2 + to 3 — n 3 + ■ ■ • • 
m 3 — n. 2 + . . . . 
not less than zero. 
The case (1 ; to : n) — (2; m ; oo) is thus completely solved by a method which 
* ‘ Proc. L.M.S..’ vol. 27. 
