sums of several classes of infinite series. 
255 
operation produces 
(log xY x + x 
28 — 1 sin 0 
1 
•r 2 -f x 
X^ + X 
2 sin 20 3 sin 30 
the value of the constant c, which is equal to the series 
c==2 . 1 
C 1 sin 0 2 sin 20 
1 
J * — &C. } 
1 3 sin 38 3 
cannot be determined from this equation, but by a second 
dx 
x 
multiplication by — and again integrating, it may readily be 
found : this second operation gives 
(log x y log* c , c 
1 1 1 i 2 si 
—I 
X X 
X m X 
+ 
• — &c. 
2.3.0* 1 1*2 i a sin 0 2 2 sin 20 ~T~ s i n jQ 
If x = 1 , c = 0 , put x = cos 0 -{- 1 / — 1 sin 9, then we have 
(0 v/_ 
f < =»✓-* SH+r fc l = 2 ' / - 1 
* 
2.3 
From this equation the value of c may be found ; it is 
, 1 
1 f 0 2 , c ±1 ■) 
Tl 6- + 2S “ } 
1 J 
The value of c thus found, we have the series 
0 (. 6 * z 2 j 1 sin £ 
(log xy 
2 . 0 
and 
(log x ) 3 , log x f 8 2 , _ e d:i') x — j? 1 
, - 2 -3 
£ +2E . £*+£ 
2 sin 20 ~r 3 sin 30 
&c. (5) 
+ f 1 o — 1 • 
0|6 " j 2 3 i 2 sin0 2 1 sin 20 1 3 2 sin 38 
In the first of these put x = cos 9 -{- — 1 sin 9, and it be- 
+ (6) 
6 i I o ± I cot 
comes — -7- 4- — S = 
6 1 0 Z 2 I 
cot 20 . cot 30 
— &c. (7) 
* Throughout the course of this Paper I shall have continual occasion to employ 
the series — — -f -h — & c . ; they can always be expressed by means of the num- 
1 2 3~ 7 
bers ofBERNOuiLLi, and tile powers of x, and for the sake of brevity I shall always 
"4** j 
denote them by S , 
J jin 
