sums of several classes of infinite series. 
2 59 
And since A ( — 1)* v 21z = — (— i) z v ZIZ + 21 — (— i)* v 2lz = 
— (y 21 -j- i ) (— i ) z v™ we find 
S( — l)* v 2iz = 
w 4i + i 
Integrating each term separately, we have 
^fr-=-{A^ + A^ + A^ + fc! 
1 2 3 
Let this integration be repeated n times, it will give 
(— i)*4'»“*=A 
v* x (— i) 
(W*+I)" 
+ A 
»**( — i) 
(v*+iy 
; A v %=± r + &c. 
(ft°-f- l) n 1 
Let v = cos $ V— i sin 9, and z -f ~ for z ; this becomes 
(^.a)*S"(— 1 )‘ 
““ ^V 2z + n — 
(-O' 1 
2 (cos 20)" 
+ fl^VF + &c d 
And finally, 
( — S) n ( 2 ) ^ + A (cos 0)” A 7 c°s A ( C os 30 ) n 
1 2 3 
X •Xfi 
== A (COS 0) n A (COS 20)” “l" A (cos 30) n ^ C - ( 1 >s) 
The integrations here indicated will, as in a former instance, 
generally surpass the powers of analysis in its present state ; 
but a contrivance similar to that which has been already 
stated, will in many cases elude the difficulty: the artifice 
consists in investigating another similar series arranged ac- 
cording to the descending powers of the variable, integrating 
it in the same manner as we have that marked ( 1 , 3 ), and 
adding these two results, we shall in many cases have a func- 
tion which is integrable, and the two series become equal in 
the case of 1 = 1 . By commencing with the descending 
+ 
