OF THE COMPOSITIONS OF NUMBERS. 
879 
and the relations 
-^<1 + 
-^K. + 
^K. . + 2a« 
7-1 
A. -1- 2a^ 
= A,^ + 
= 2 (A^j + aj 
hence the numerator factor becomes 
2 (A^^ + “-Ki) (A^, + . . . (A^_ + “k,) — (^k, + • • • (A^^ + 2 (A^^ + = 0. 
53, This result proves incidentally, as stated above, that the final fraction 
^Ki ^K2 * * ■ ^Kn A^^ A/cg • • • A;^^ ^2 • * ■ 
(1 - Si) (1 _ S,) ... (1 - S,d 
vanishes identically. 
It also terminates the proof which it has been the object of this portion of the 
investigation to set forth, 
54. The analytical result may be stated as follows :— 
The fraction 
1_1_ 
^ {1 —§1 (2«i + «2 A • • • "t *«)} —% (2«i + 2a2 + . . . + a„)} ... {1 —Sli (2«i + 2«o + ... + ‘io-rM 
is equal to the product of the fraction 
^ 1 — 2 (2 Si«i — 2 s-^s^ci-^ci^ + .. . + ( —)“+^ s^.So . . . s,i ai«2. . . «„) 
and the series 
I + s 
where 
2 (A „^ + (Ak^ + i^Ka) • • • (Ak, + ci^) — (A,,^ + 2«^^) + 2«^J .. . + 2«^p 
(1 - SJ (1 - S,,) . . . (1 - S,,) 
[2ciy + . . . -j; 2a^ + «K + i + . • • + ««) = (A^ + 2(x^) 
and the summation is in regard to every selection of t integers from the series 
1, 2, 3, . . . 71, and t takes all values from 1 to r — I. 
55. It may be interesting to give the simplest cases at length. 
Order 2. 
1_ \ ___ 
^ (1 — §1 (2ai + «2)} {1 — So (2«i + Saj)} 
— 1 
— 2 
1—2 (Si«i + .S2«2 ~ SiSja^aj) 
X 
1 + 
■Si«2 
+ 
^2^1 
1 — §1 ( 2 ai + * 3 ) 1 — S^ ( 2 ne, + 2 « 3 )_ 
