H4> 
Mr. Woodhouse on the Integration 
(for e' being 
IS- 
I — y/(!— £ 
[i + e] 
f — Ew'-f 
+V/4-- 
+ &c . } 7" 
XT t a e 2 . e'. e". &c. .. 
Now, -7 = 
’ P 4 -4 -4 
.(i+0(i+O(i4O -.('40; 
consequently, since <? 7 . e", e'", Sec. continually decrease, the quantity 
S 
cU may be rejected, and = dv 
[i-e z V 2 ) ■ 
pj v “-v J J v — V(i-v 
nearly = ; consequently, 
/=^( 1 + 0 «' 4 fr^ , - ±£U ^“'' 4 &c. ( 5 ) 
I p r dv P r e z e z .e' . e z e‘ . e" . « \ /V* 
7 ” J~\/[ l — V z 2 n {_ 2 2 . 2 2.2.Z ~t" • J J 
When a: passes from o to 1, u ' passes from o to 1, (its maxi- 
mum,) and from 1 to o ; similarly, when u' passes from o to 1, 
u" passes from o to 1, (its maximum,) and from 1 to o. Hence, 
generated from x = o to x = 1 = 2 fv{'-S z ) > from u '—° 
to «'= 1 ; = ’ from = 0 10 1 ; = r Svr=^v 
from v — o to v — 1 ; consequently, since u', u ", u"\ Sec. = o, 
when x —i, the whole integral of dx J from ^ = 0 to 
x — 1 
= . JL — L{ fl 4- — 4- g/ ~ A + Sec. T 2”. 
2 n 2 2” L 2 1 2.2 1 2.2,2 1 J 2 
= (putting Q = i ^ + &c.) P . i - PeQ i 
r=P(l- C Q)+ (6 ) 
Which is the same form as was first given by Mr. Wallace, 
in the Edinburgh Transactions, Vol. V. p. 280. 
The form (5) may easily be made to agree with that given, by 
the last mentioned author, for the length of an elliptic arc. 
