. 92 
MAJOR P. A. MacMAHON: SEVENTH MEMOIR ON 
F ? (3m; 13) = F 9 (3m + l; 13) = F ? (3m + 2; 13) = m+1, 
F g (3m; 1 2 3) = |-(m+l)(3m + 2), F g (3m+1; 1 2 3) = \ (m + l) (3m + 4), 
F ? (3m + 2; 1 2 3) = f- (m+ 1) (3m+6), 
F ? (2m; 1*’2* 2 ) 
_ /m -\-ho — 1 \ /A/j +1\ /m -\-lc 2 —2\ + 3\ /m + —3\ / 2 m + *i — 1 \ 
A 4-1 / + \4-i/\ 4-1 / U.-iA 4-1 / + - + l v 4-1 ;■ 
F jr (2w+1; 1*'2* ! ) 
_ l Tc l \ (m + h 2 —1\ (k 1 + 2\ {m+Jc 2 — 2\ (2m + h\ 
~\K-V\ 4-1 / + \4-i/\ 4-1 y + - + U>-i/’ 
F ? (3m ; 1 A '‘3 A ' 3 ) 
_(m + k 3 — 1\ fk 1 + 2\ [m + k 3 — 2\ /^-t-5\ [m + lc 3 — 3\ , [Sm + Jc 1 — 1\ 
A 4-i / + V4-i/V 4-i / + \4-i/\ 4-i A"'A 4-i )’ 
F 9 (3m + 1; H3* 5 ) 
_/ \fm + k 3 — 1\ (k l + 3\ [m + Jc 3 — 2\ (Bm + kf 
"V4-1/1 4-i / \4-V\ 4-i / \4-ij’ 
F 9 (3m + 2; l*’3* s ) 
_ /^ 1 + 1 \ fm-\-k 3 — 1 \ + 4\ /m + & 3 — 2\ /3m + ^ +1 \ 
“U-l/V *3-1 / Ul-l/\ *3-1 / + \ *1-1 /' 
The use of the Hammond operator D m is convenient but not essential to this 
investigation. It is convenient from the algebraic point of view, and also because 
it brings into prominence properties of the operator which are in themselves 
important. The coefficient of a-T'af” 2 ... a s m ‘ in the product Q/'QA... Q/“ is readily 
obtained wTien we remember that 
Q — _i_ 
(WHi-^MW). 
and the various modifications are readily made for the allied functions A„ B„ ... U t . 
The Partitions of Multipartite Numbers into Two Parts. 
Art. 11. The generating function which enumerates the partitions into two or 
fewer parts is 
i(Q, 2 + Q 2 ); 
and since, from the principles just stated, 
D 2 A a = ( 2 m + l)v, D S „ + 1 Q , 2 = (2m+2) Q,, 1 , 
^2m0>2 = Qa> I^2>n + lQi2 = 0 > 
