sums of several classes of infinite series. 263 
In order to integrate this equation, let us suppose c zk n to 
f/ 
represent the co-efficient of r 2k in the developement of — 
where a (r) — 1 -j- Ar* -j- B r 4 -|- Cr -j- &c. 
f (r) 
- = 1 4 -re 2 -I- r*c + &c. 
n 1 z,n 1 4 ,n 1 6 >n * 
then . . 
{a.r) n * z,n 4 
If this be multiplied by ar, it becomes . 
— - - r ^- = 1 4- fc 4- r 4 c 4- r a c, -f- 
( ar ) n — 1 1 2 > n 1 4> w 
r 2 A 4- r*Ac 4- r s Ar 
1 *-» * 4, TO 
2, TO 
r 4 B + r‘Br 
+ r 6 C 
But the co-efficient of r 2 * in this series is equal to c ik h __ i 
hence 
Ac 2^-2,TO+ BC 2^- 4 ,to+ &C - = ^,TO-I 
This equation will become the one in question, if we make 
A = B = &=» C = (8 /--> 6 &c. 
1.2 1.2. 3.4 1. 2.. 
This produces 
* -I- (V-0* g 4. 4-&c -r 
1.2 2* — 2,toT- 1. 2.3.4 2^ — 4 , nT <XU ~ 2k,n — l 
We have now only to determine the form of f(r), and this 
may be easily accomplished since the values of are known ; 
for if we put n = 0 
f(r) = 1 4 - r‘c 2 a + r‘c^ o + r\ o + See. 
= 1 + 2)’S ~ 4 - 2r*s + 2/S^i 4 - &c. 
Therefore c 2k n is equal to the co-efficient of r ik in the deve- 
lopement of 
~+~ t +1 
1 + 2r*S — + 2r 4 S — 4- Sec. 
I l 4 
(cos rb) n 
Or if (cos r&)~ n = 1 + A 0 2 r 2 -f- B' 6V 3 + C' 6V+ &c. then 
