of certain differential Expressions, &c. 
229 
P ut_ J~r = c, and dx J = 4 f> 
then df=\J + + + 
Now (by methods similar to those that have been given) 
,I— 1 . in — iv 7 ” — J 
1 + 
&e. } 
3. ...2 « — 1 
2 . 4-...2K 
+ 
2W+I 1 (2«+l) (2«— I) 
-t/ 2 ) 
+ *}' 
rj/pfh_ __ > rs 1 ”- 1 , 
J\/(l — V Z ) ’ v (t ^ J p 2?i ‘ 2«.(2«— 2) 
f£=, and [£%£ = 
i-/ *v/ 1— z; 2, V ( 1 — t/ a ) ' \ 2 
+ &C. _{_ 1 • 2 • 4 2 ” 1 , 
1 1 1.3.5 •••• Z«+I j V * 
Hence, putting <p' = f = ± r _ . ) 
Difrv/ (1— z) 2 ) +D 2 if c 2 { 
/=lv/ “j _+ 4 ■ 74 -"-) + H}+ & ' | 
which series agrees exactly with Legendre’s, given in Mem . de 
VAcad. p. 620, when the quantities v, V 1— z> 2 , &c. are expressed 
in geometrical language. 
In order to find the integral from x — o to x — 1, put x = o, 
then v—i, put x — -^==, and then v — o; but it has appeared 
that the (between the values of x, o and 1) — 2 
(between the values of x, o and ~=j = consequently, 9 . J 
(between the values of v, 1 and o). 
Hence, ,/== J (2=£) i+D- ti. | + D 4 lf.^ . + &c. | ( 2 ) 
or= \/(i — — — { 1 — . \ . c 7 — 1 - ■ ~- 5 . c 4 — &c. ) 
V '- 2 2 i 2-4 2 2.4.6.82.4 / 
which is the series given by Legendre, and by Euler, Novi 
Comm. Petrop. Tom. XVIII. p. 71, and called by that author 
Series maxume convergens ; yet the series is by no means prac- 
tically commodious when e is nearly 1. 
A very useful series, when e is Small, was given by Mr. Ivory, 
