476 ON THE SUMMATION OF SERIES, 
sum of v terms of the series, whose general term 
is Ho(w+hyt, is made to depend upon the inte- 
gration of f(@#)da, and the derived functions of 
f(ox). Again, since equation (8) is true for any 
value of x, it will be true when v=y; then we 
shall have 
¥ 
ae F io iv+t)}=| tou)d + (A+h’) f (ey) + 
I 
2 I 2 3 II 
(B +Ah’+ f (2) +{ Bai +Ah’ + fh’ \f (ay) 
2 2 2.3 
I 3 
+ (D+ BW pa” + 10°) (ey) + (Di! +80 + 
° 2.3 2.3.4 23 
2.3.4 2.3.4.5. 
eee ol )f (ey) Hor gh tities ae (9) 
And if we subtract equation (9) from equation 
(8), we shall have, 
z. f} ®(a +h) : = he f (or) dx + (A +h’) 
y a 
{f(ox)—HeyR+( BAN +0" ff (ox) -£ (ayh 
+( Bh’ +AN/ +20 \S (ox) —F (ey) +(D+B0 + 
AN 4 wf f(x) -f (ey) + Rosismayenin (10) 
2.3 2.3.4 
