382 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
Fig. 20. 
by two confocal spherical parabolas, having one common focus, and the nearer vertex 
of the one curve on the focus of the other. Thus 
let F be the pole of the hemisphere ABD. Let BC/* 
and AC^F, denote two spherical parabolas having 
one common focus atF; F^ and/ being the other foci. 
Let Yf=<y, and therefore FF= 2 — y. Hence the mo- 
if we make cos y—l, cos y^=/ 
Thus while the arc of the one is given by the 
integr 
sin^cp 
the arc of the other depends 
on the integral 
LVII. On the value of the complete elliptic integral of the third order and 
logarithmic form. 
Let 
r 
dip 
[1— nsin^p] a/i — i2sin2,p 
Assume k the criterion of sphericity=(l — — 1^ 
then 
dp 
■Jo N^r 
(320.) 
A 1 If 
d«[_Jo N-/IJ n}. 
1 f 1 dp 1 r 
*2 dp 
'n}, NWT nj, 
oNv/r • • ' 
2>cf'2 dp 
2xCt dp 
J-jzJo WVl 
^}o Nv/I • • 
p. 195) 
pi" dp /^^ — n'' 
vfl- dp /t 
lo N^I 1 . 
a ^r“Jo 
(321.) 
(322.) 
(323.) 
and 
HCi dp r2i^ 2 A!_lo10 
NV'l~ n « 
■ dp 
nVj- 
Introducing the substitutions suggested by the two latter equations into (322.), 
2z 
dK 
QX / \ 
Now 
