228 
Mr. Woodhouse on the Integration 
3.1 .6 / — *V( 1 — **) 
&C. 
6 . 4 
5 • 3 • * ^(i— * a ) 
6.4.2 
i 1 • 3-5-9 \ 
' 2 . 4.6 } 
Hence, if the integral of dx. J ), between the values of 
x, o and 1, be required, putting = ■ — = value of <p, or of 
fjTi-^Y when x = l, we have 
jdx j | — (from jt = o to a: = 1) 
w r 
4 1 c 2.4 C ~ ' 2 . 4 
or, developing the symbolical coefficients d it, d* it, &c. 
— d 3 1 2 * 6 
2.4 c 
^TT + Mf 1 ) 
2 ^ ^ 2 . 2 ^ Z . 4 ’ 2.4 
3 * 5 
, . . ~ — See. } 
2 . 4.6 2 . 4.6 j 
which series has been given by several authors, Simpson, Euler, 
Animadversiones in Red. Ellips. p. 129, &c. 
If, instead of the coefficients 
2 - 2 . 4 : 
, we use — di~*2,d 2 1—5 
&c. the integral 
= ~{ 1 + D1 ^- D 1 “*t £ 2 + B 2 if. Y 1”^.^ 4 +D 3 li.D 3 l - ^.^ 6 + &C.| 
where the (ft -j- i)th term is D” if. D" i~i, which, (since 
Y i — ^ — — d” if. (2ft — 1)), equals — (D” lfj* . (aft — 1) ; con- 
sequently, the integral may be put 
f { 1 — ( Dli )‘ • - (?* !<)*.• (?’ **)* ' & - &C ‘ } 
From this series, Jdx J ( ' ) ma y be computed when <r 
is small ; but it is evidently of very little use when e is either 
nearly =1, or is of mean value. To speak in geometrical 
language, the length of an ellipse of small excentricity may be 
-dv 
computed by the above series. 
If V be put = 1 - sx*, £j— r) _ i), 
and^y(^)==^ r .y^v(i+^). 
