OF THE COMPOSITIONS OF NUMBERS. 
871 
37 . The whole coefficient of being 
^0 "h + 02^^ + . . . + + • . . 
a=: S + 
^21 
P 
§21 + S 31 
’32 
2 + Sc 
'31 
^21 + hiJ \hi + 
‘’31 
the summation being subject to the condition 
•^21 + “h hi — 
Moreover, denoting the whole number of compositions of the tripartite pqr by 
F we have 
F (pgr) = S C, 
_ 2P+i+r-l ^ 
38. Hence F (jigr) is the coefficient of ctP^iy^ in the development of the product 
2P+2+-,-i (a + p + ly)P (a + /3 + hY (« + ^ + 7)h 
which is more conveniently written 
1 (2« + ^ + y)^ (2a + 2^ + y)^ (2a + 2,8 + 2y)'-. 
39. This product is a generating function which enumerates the compositions of the 
single multipartite number pqr, but the generating function of all trij)artite numbers 
can be at once derived from it. 
It is 
1 _I_ 
2 {I - S(2a + ,3 + y)} (I - ^(2« + 2/3 + 7 )} {1 - «(2« + 2,8 + 27 )} ’ 
in which, when expanded, the coefficient of {sa)p{t^y{iiyY is the number of com¬ 
positions of the tripartite number pqr. 
The generating function previously obtained for tripartite numbers from the 
analytical theory was 
« + y8 + 7 — ,87 — 'ya. — Ci^ 
1 — 2 (« + y84-7 — /^7 — 72 ^ — «/S + a/Sy) ’ 
the number of compositions being given by the coefficient of a.P^'^y', The addition of 
\ to this fraction brings it to the better form 
— 2(« F/S-f-y — /Sy — yo. — «/3 + aySy) ’ 
