262 Mr. Babbage’s new methods of investigating the 
0 1 
— 4 • c = 
7 8 7.11 
\- &c. = c 
s ?z— I 1 2,n- 
i (cos 0 y 1 - 1 2 * (cos 6) f 
The equation c z n — c z n _ l = ~ being integrated, gives 
. n 0* , 7 
c — — — 4- b 
2,71 2 2 1 
If n = o 
c 9 =6=^-4-+-, — &c. = aS^-' 
2,0 I 1 2 s > , 1 
It I 
i 1 
Hence 
flog. a?) 2 
2,71 
— + 2 S — ■ and 
2 1 /* 
. 71 0 1 , o ± i 
+ - • — h sS — : 
1 I 2 1 Z l 
1.2 
1 
jr + x' 
i*(cos 9) n 
s* + + x 3 o , . 
2 1 (C0S 20)' 1 j 2 fcOS30)” v ) 
These integrations being repeated, we shall arrive at the two 
following expressions: 
_i_ Oog *) 2k - 2 c . &r . (i°g*) 2 r , , . , __ 
1.2... 2k * J . 2 , , 2k — 2 2>n T“ MC * i — C z k - 2,n -h Czk,n — 
- X + X 
, — I 
x 2 + x~ z 
r 3 _L jr 3 
L & c 
I Z *(C0S 0) n 2 2 ^( COS 20)” ~ 3 2 *(cos 3 0) n ~ 
+ J! 2 £il!!l! c + &c .c. = 
1 . 2 .. 2 A +1 1 1 . 2 . . 2 * — I 2 ,n 1 2 ^,n 
( 2 >3) 
x “~ x - UZ1 L - * a " gr C 
J 2 ^ + ! (COS0) M 2 2 * +, (C0S 20)” 3 Z ^+ ^cos 0) n 
It now becomes necessary to determine the value of c zkn , 
v\ hich is equal to twice the sum of the series we are investi- 
gating ; for if x — 1 
2 —2 
X X 
r 3 _ x 3 
2 k> n i zA (cos 0 )” 2 2 \cos 20)” 3 2 ^(cos 30)” 
For this purpose put in the first of the equations ( 2 , 3 ) x=. 
cos 0 + t/-i sin 0 , then the series on the right hand is equal 
to c zkn __ l 7 and we have for determining c zkn the equation of 
finite differences. 
<*£=£ + 1 , (v-.^r 4 , +&C. + C, = *. 
r |. 2 .. 2k— -2 2,71 * l.2..2k — 4 4,« 1 1 2R,7l 2R, Tl— I 
— &c. 
1.2 ... , 2 k 
