of certain differential Expressions , See 
2 . Let 2 m=^ 
rdd , 2— e 2 rdb 1 rdb 
J Rs 3 * 'J R3 ~ 3 (i_ e 2 ) J-R 
263 
V( !-■**) 
2 _ 2 -g 
3‘(l_e 2 ) 
g* 
3(i_ e 2 )- X R 3 
■ry/(I-^) 
_ r J _ A 2 ~ g *v(*- 
z J 3 • (i-O J R 3 ' * R 
R 3 
3 . (1 — e 2 ) 
if x = 1, 
/ 4 -= 4 .^./( 0 - T ^r/^- (/) 
This is as commodious a form as any for computation, but it may 
easily be changed into others; thus, since J-~=f — e . 
fw- = f • /( 1 ) + TITZTJ ■ % ’ (integral taken from 
x = o to x = 1 ) ; 
f dd r dx i + e' r du’ 
or, sine eJ- T -—J-^j ZxZ){i _ e ,^)— 2 V{l _ u '* ) 
2/(0 
737 - —/t 1 ) -TZ 7 ’ b y e q uation (*)» P a g e 240* 
. __ 5 + 3 gl2, / x + g' \* r r \ 2 . (i+eQ 1 r,f \ 
J R s — 3 . (i-O 2- [i-e'l J v 3 {i-e'y ’J V >* 
or 
5 + 3 g * ./(O , > I /Vl) 
. (i_e ') 2 ’ j — e 2 3 ‘ (1 — e 2 ) (i— e') 'J v /* 
/<») 
3- (! 
the integral being taken from x — o to x=i. 
The integration of the form - d _f x ^ ( 1 — g 2 x* depends 
also on the integration of R v ^_ x , } and of dx J | 'v^r )‘; 
for, substituting as before, and taking the differential of 
x zn ~ l v/(i —r) • R 2m +\ (X), we have 
dX-- 
(2m + 2n) c a — .(201+1) j: 2 "— 2 R 2 ^ 1 . 
-(2«4 _2;w + 1 ) *^ 2 ”- R 2W,+I .</0 
+ (2m + i)±=£.x zn - 2 .R^-'.dd. 
Hence, /*» . R™+ 1 . <rt = -- ^"11^/ ' •/*““ 2 • R 2 " !+ '- <« 
mdccciv. M m 
(s) 
