230 Mr. Woodhouse on the Integration 
in the Edinburgh Transactions, Vol. IV. which I shall notice in 
the sequel ; not now, because I consider it as a particular case 
of the general method by which, in all cases, the integral 
of dx J may be computed. 
In order to deduce the series by which, when e is nearly = 1 , 
/ may be computed, put 1 — e 2 = b% 
thendf=d.ry(i+ rdM = (ifx= v(1 ,.*.)) 
b* z* dz 
Now, 
Vi 
V((I +* a ) (-i+A a * a U. V p +* -) l 1 1 c j 
b 1 - z x dz 
and the (;z-j-i)th term 
Now 
z ln + z . dz 
rz *»+ 
J V(i 
+ zi ) 
2 a ” +1 V(I+Z*) 
„ i . z 2 ”+ 2 ’. dz 
** B+1 ^\ 1+Z% L _ zn + l _ C zXn dz . and, consequently, 
+ 2 2« + 2 J V(l+Z*l ^ 
z 1 "—* dz 
2» + 
2«-f I 
2M+2 272 + 2 . 272 
272+ 1 .Z 
„2»-I Jfi.MU (272+l)(272 -l) f 
.% V[l-r z )-r {zn + z) zn J 
V'l + Z 1 
- _L 2 «+0( 2 «r_li . % 2M — 3 _L &c. 1 
■ ( 2?2 -f- 2 ) Z 72 . (272 — 2 ) 1 J 
V(l+2 2 
\/ ){ 2 /l + 2 272 + Z . zn 
t (272 + l) (272— l) • • • • • 3 • 1 
( zn -j- 2 ) 4 
Hence, 
,7 (l+Z*}* 
=7(TT?r v/ ( 1 + i ’ 5; ‘ ) 
— {Dl-^ + Dl-iD‘ l“i 6 4 +D* 1 D 3 1-4 S + D 3 1-5 D 4 ,l-i6 8 +&C.| 
x log* (z+V(i+^) 
„ . _fVVL±£lL 
— di -HV( i+ z *){ 
3* 
i-i&V(i+2-){ 
Z 5 
4.2 
5 z 3 
f - &c. ) 
* 6 . 4.2 J 
6 . 4 
— &C. 
or, since -/ ( i+z a ) = 
