41] 
267 
41. 
ON CERTAIN FORMULAE FOR DIFFERENTIATION WITH APPLI 
CATIONS TO THE EVALUATION OF DEFINITE INTEGRALS. 
[From the Cambridge and Dublin Mathematical Journal, vol. n. (1847), pp. 122—128.] 
In attempting to investigate a formula in the theory of multiple definite integrals 
(which will be noticed in the sequel), I was led to the question of determining the 
(i + l) th differential coefficient of the 2t th power of J(x + X) — J(x + g) ; the only way 
that occurred for effecting this was to find the successive differential coefficients of 
this quantity, which may be effected as follows. Assume 
Uk t i = {{x + X) (x + /¿)P y(œ + X) — V(® + g)Y\ 
then 
iJjcjdx w (x + \)(x + g) \/{{x + A) (x + g)} 
= P 
{V(æ + À,) + \/(x + g)} 2 — 2 v /{(æ + À.) (x + g)} 
(x + X) (x + g) 
V(x + //-)} 
% 
or, attending to the signification of U k g, 
Hence 
&c. 
34—2