OP THE COMPOSITIONS OF NUMBERS. 
869 
I'he p — S .21 — 532 letters a may be distributed 111 + 5 ^ + 1 positions in 
P 
%1 + ^31 
ways. 
The q — S .21 — letters /3 may be distributed in 531 + .<? 3.3 + Sgj — cr + 1 
positions in 
1 + 
%1 + ^32 "t ^31 — ^ 
ways. 
The r — ~ letters y may be distributed in S 33 + 53^+1 positions in 
/ 7’ \ 
[s+s 
\*.32 ‘ * 31 / 
Hence, for any given permutation of the terms in 
{I3ay--’^ (r/3y--" (y/3ar (r«) 
S 31 
there are 
P ] ( ^1 + hi W ’’ 
,^"21 "t ^31/ W 2 I + ^32 "t S 31 aj ySgj + Sgi 
arrangements of the remaining letters which do not introduce additional /3a, y^ or 
ya contacts. 
Hence for a given value of o- there are 
P 
2 + s. 
'31 
(Sgi + %2 "h ^31 — ^) • 
% "t ^ 31 / \% "8^32 "t % V%2 "t %/ ®’) • (®32 *^)* ^ 
permutations. 
To complete the enumeration we have to sum this expression in regard to o, 
We have 
2 + s. 
'31 
(^21 "t S 32 '^) • 
'% S' ^32 "t S32 (Tj (Sg;^ <j) ! (.Sg.j tr) ! (7 ! Sgi ! 
= X 
(2 + %)! 
(2 - *21 - *32 + ■ (*21 - O') ■ (*32 - 0 -) ! O- ! S3I ! 
Soo! 
(2 - *32) ! 
_ (2 + S31)! V__ 
*31 • *32 • I2 ~ *32) • O’ ! (S32 cr) ! (Sgj O') ! (2 *21 *32 "t" ' 
_ (2 + * 3 l) • ^ /* 32 \ /2 *32 
*31 • *32 • (2 — *32) i W / \*21 ~ O' 
_ (2 + * 3 l) •' / 2 
*31 • *32 • (2 ““ *32) • \*21 
_ /*21 +S3l\ /2\ /2 S *; 
‘’21 
*31 
* 32 / \*21 “t *31 
