I 
i e _ x ^ 
2.3* R 3 "^2.3. 4*5 rs 
Infinite Series • 
3* 
Ex. 2 . Log. (*=i=0=y^= log- **;=*?:£; 
fis-{ X +"e? ~ f 7=7 + 4W + &C> 
Ex. 3 . Z' — — ~ — r- a = P -p—z ~ t ~- 2 + &c. 
Ex. 4- P -7-4 rr, = P-P=*=~r±Scc. 
* J V / (I”0 , ±<) ) </ V" X -j 2 
Ex : * / 7rir-= fvh-fp + fpk' 
io 9 
See, 
Ex. 6. Let the fluxion be 
*7 
, where c is a very final! 
quantity ; then, if P be put for \/ 1 — x z 9 the fluxion becomes 
- &c. of which the fluents will be found 
SC-* 5 P 
2 . 2 A • 7 b • 
x CX X C X 4 X e 5 x X 
2P 
8P 
i6P 
A— - x 
2 
C I . A-*P c z w 3B X 3 ? 
~8 X 
4 
— — X 
16 
&c. where A; 
J*L- t B=r^-- P ,c=^=— , &c. 
This is the fwifteft converging feries for finding the length 
of the arc of an ellipfe nearly circular, which is yet known ; 
for example, let the abfeifs to the axis beginning from the 
center = x 9 the femi-tranfverfe axis of the ellipfe be 1, 
its fembconjugate i-d; then will c~2d— d z 9 and let the 
length of the quadrant of the ellipfe be required, in this cafe 
1, and P = v/i — # z = o ; and A = = 1,57079,, 
&c. whence the length required is 1957079, &c. x (r — 
2 . 2 
