234 
Mr. Woodhouse on the Integration 
Let y = l ( 1 + 1 )» then* dy 
dx dx 
dx 
2 y/[l + x) (1 + ^( 1 +*)) 
dx dx f —i- 
2X\Z[l+x) 
1 
2X V 
x J . 3 
t 2.4 
- + - 
* -7. 
3 
2 • 4 
1 - 3-5 
when x = o y = /2 = 
hence, / 
2.46 
corr. 
— &c. } 
■ „ ~ &c. -f corn 
Vb 
— 1 • 3 b * 1 
4 2 . 4 * 4 "r 
-L-t+tt 
=-j“ ^2 -j- l 
V[ 1 + 
i-3 5 
2.4.6 
4 2 
6 3 
“ 6 
3 • 5 
VP 
l _? * l I • * __ L- 5 
v/6 
-144-- 
24 
+ &c. 
6 3 + &c. } 
I 3-5-7 
4.6 
If this series be substituted for / i±Ljl±^l 
V» 
— Sec. 
, in the above 
V(i -\-b) 
form for/(i), if - J - y— be expanded, and the terms affected with 
like powers of b, be collected, we shall have the same series as 
Legendre has given. Euler, however, is the original author of 
the series ; and has expressed its law much more clearly than the 
French mathematician. In the Euleri Opuscula , Berlin, 1750, 
p. 165, the author says, that the elliptic quadrant 
= 1 + A&’ 4 -B& 4 + Cb 6 + &C. 
— { ah * -f /G6 4 ~|- yb 6 -j- See. | log. b, in which 
A = log. 
B 
■A-H— j3)+J-. 
C = 44 r B - #-?)+*■ 
2 . 4 
D 
E = -|^_D 
&C. 
*(*-•) + *-t 
,G = . fl 
H 2.4 
3 '5 /3 
7 = 
t = 
Sec . 
4 6 
i_ 7 _. 
6.8 - 
7 - 9 
8 . 10 
