478 ON THE SUMMATION OF SERIES, 
series whose general term is f ‘0 (@+h) t, first 
when wv is continuous, that is, when w takes every 
possible value between o and w; and secondly, 
when @ is discontinuous, that is, when w takes 
the values 1, 2, 3, &c., respectively, we shall 
readily see from equation (11) that the difference 
of the summation of these two series will be ex- 
pressed by 
o(a)=( A+’) (ow) —f (ay) t+(B+An'+ i ) 
I I 2 3\\ cot lee It 
Jf (00) —f (ay) + BA'+ AW 4 2 }}£ (00) -f 
{ 2 2.3 
2 3 4 lil III 
(ey)k +[D+BH +Ah’ + hi \ff (ov) —f (ay) t+ 
> 2.3 2.3.4 
n—I n—I 
whose general term is 3 (h')} f (ox) —f (ay) 
oo n—t1 n—tI 
so(@)=E. 3 (K) IF (ou)-f  (ay)h 
] 
therefore we shall have from equation (10) 
x x ca n—t 
{ £(ov)du== . ffo(w+h)t—3". 3 (h’). if 
1 
¥ g 
Geis) sate acoy) be oid dteen ah 0 (13) 
