OF THE COMPOSITIONS OP NUMBERS. 
847 
Substituting in a previous identity 
C I ~ C - s) ~ • {PilhPz • • •). 
therefore, 
i C ^ s ~ ^) C - 6’) ^f^P^P^P^ . . .,r- ii + s) . . . .) 
= ifiPiVzPz • ■ ■ P,r) . {pMh • • •)• 
This is an absolute identity and, equating coefficients after writing 
= </> {r, s), 
+ s — 1' 
s J \fl — SJ 
we obtain 
p)f{PiPdh . . ., ^■) + {r, p - l)f{piPdh . .., r - 1) + .. . 
+ (j) (r, p-r+ ^)f{piP.2P3 .••,!) =f{PxPdh ‘ '>')■ 
This formula enables the calculation of the number f {piP^Pz - p} ^’) fbe 
successive numbers 
fiPiPiPs - r), fip^PzPs..., r- 1), ...f{pi2W3- !)• 
14. A more useful result is obtained by summing each side of this identity for a 
values of r from 1 to S p + /r. This result is 
r 
^ p) + (}^{r + 1, p - 1) + ... + (l>{r + p, 0)}f{p^2hP3 ... r) = F {p^p^Ps ... /a). 
It can be shown that the expression in brackets [ } to the left has the values 
2 '"' 
/r + p — 1\ 2r + p ^ 
therefore, 
P 
F {PiPiPz • • • /^) = ^ 2'^-' ^ f {pMo^ . .., r) 
= 2^-'(p, + ^)f{pxPiPz •••.■!) + 2 ^^-^{PiPm 
+ 2-^ 3) + . . ., 
•. 2 ) 
a formula of great service when p is large, as the number of arithmetical operations is 
comparatively small. 
