12 
MR. TALBOT’S RESEARCHES IN THE INTEGRAL CALCULUS. 
Now these three values satisfy the theorem ; for we have 
arc sin -r = 53° 8' 
o 
12 
arc sin = 67° 23' = & 
56 
arc sin — = — 120° 31' = f 
and 
+ 0 + &' = 0. 
§ 5. Application to the Ellipse. 
In order to obtain a relation between three elliptic integrals, the simplest suppo- 
sition which we can make appears to be 
/ 1 — e 2 x 1 
y~r -^ =vx+ i, 
whence 
3 I 2 9 , 1 - e s — V 2 2 
x 4- — x l 4- s . x ■= 0. 
1 V 1 V 2 V 
2 
This determines the value of the symmetrical v to be = — : and therefore making 
this substitution we have 
" 3 + r x 2 + ( — 4 C - . r 2 — 1^ a; — r = 0 
x 
and 
whence 
But since 
/ 1 — e 1 x 1 2 
V 1 -**■ - T * + l > 
/ 1 . /?2 O 
S d x \ / — — = — 
V 1 — x r 1 
S x 2 = p 2 — 2 5 = r 2 — ^ 1 ( , e . r 2 — 2^ = — 
l + 
S # c? a? = — - — r d r 
+ e 
. r 2 + 2 
Also 
— S x dx — (1 -j - e 2 ) dr. 
S d x — — dr 
2 
:.—$>xdx-\-$>dx = e 2 dr 
■•■S dxy/ '~**! = e 2 dr 
= e *r + C. 
