AND ON DEFINITE INTEGRATION. 467 
consists in the determination of F(#+1)—F(2r)= 
(o(x+h+1)}, by means of its successive differ- 
ential equations, and then eliminating from them, 
by Lagrange’s method of multipliers, the functions 
II. II. EV’. 
F(z), F (2), F (x), &., &e. 
Pros.—To find the sum of x terms of the 
series whose general term is fio v +hy, from the 
limits s=1 to r=2z, where hi i a ape he: quantity. 
Now, if agreeably to the notation which is 
adopted by the writers on Definite Integra!-, 
we denote the sum of the series in question by 
x 
St Jo(a + hyt, which is a function of «, say F(x) ; 
1 
then we shall have 
z Floe+mi=fea+m+ toa+nts tloa+n} 
= a Hea +1) t foe+n + fo(s +1) 
. floce+m +f}orth-+1 P(t 1), #a(2) 
