OF THE COMPOSITIONS OF NUxMBERS. 
849 
Either summing these identities with respect to r, or operating as before upon the 
expression 
^^{ViPdh • • ■ ■ ■)> 
we obtain the identities 
F • • • 1) 
= - {^{PiPdh ... 12) - 2F i'PdhPi • • • 2)] 
= + [F {2)ip.22h . . . 1^) “ 3F (piiJsPs . . . 21) + 3F {pi2Mh • • • 3)} 
= (_)-! S(-) 
... -1 (« + /3 + • ■ • — 1) ! g 
ul 231... 
F {PilMh • • • 2^“). 
These relations are readily verified in the particular case 
{PilMh •••!) = {!)• 
16. Another very useful result is derived from the algebraical result noticed in the 
foot-note to Art. 13. 
Since the supposition { — = Si leads to the formula 
aud 
— 6-1 (Sl — 1) . . . (i’l — p, -f 1) — A, 
H' ■ 
^ ^ Vp Vp. . . . -^2 
we reach the operator relation 
~d,{d, - 1 ). . . (d, - ^ + 1 ) = s D.-D/-. . . 
whenever, as in the present case, the operand is such that ( —)'^“^ == 
Assuming 
P d, {d, - 1), . . (d, - ^ + 1) H' = (0 H'-' (1 + H)' 
and operating with 
yL6 -H 1 
(c/^ — /x) we find 
(/X + 1) 
c?. (d, -1).., (d, - H-=;,) H--' (1 + Hr ■ 
verifying the assumption. 
MDCCCXCIII. — A. 
5 Q 
