DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 343 
but the equation of condition (109.) gives 
a-=- 
4;2 
art 
hence e^= 
In the same manner the spherical conic, whose radius is V, gives 
e^— 
e^e^=-^ - = . . 
. ( 110 .) 
tan/3 
or e’ and e'^ are reciprocal parameters. 
To compute in this case the value of the coefficient sin/S in the expression given 
in (16.) for rectification, 
tan/3 . 
Since 
[1 — siu^(p] v" 1 — sin^ip ’ 
.L 9r, X 2 • r. * 
tatf/3=^2i:^, taY\^ci=-^^2, sin/3=^, 
1 . T 1 • • tan^/3 . «^) 
we obtain by substitution, {i^sin /3= ’ 
but the equation of condition (109.) gives 
4,-2 
0*1 
7 , , - ^ , tan^/S . b\k^—d^)[k!’^-a^) 
hence j^sin“(3=- 
aWk'^ 
■k%<^' 
As this expression is symmetrical, we shall have for the spherical conic section, 
whose radius is k\ 
tan/3' . Of /X 
(I'l-) 
Hence 
tan/3 . _ tan/3' , , 
^^^*'"<3 = ^ 810 / 3 , ( 112 .) 
or the coefficients of the elliptic integrals, which determine the arcs of two spherical 
conic sections, having reciprocal parameters, are equal. 
Let z be the criterion of sphericity ; then as 
V2x 54 
— (l_,„)(l_'J = (l_e’)(l-e’)=p^. 
z=>c' (113.) 
XXXII. To determine the values of the angles K and X' wliich correspond to the 
same angle (p in the expressions for the arcs of spherical conic sections having reci- 
procal parameters. 
Since 
„ cos^« k ^ — 
COS i = ,2 A2 — 
k^ — a^ 
cos'^/3 k^ — 6^ k^—a^-\-a^? 
Introducing the equation of condition {k ^ we get cos£=^,; but 
yfc' k 
tan(p=cos£ tanX, as in (39.) ; hence tanX=-tan^, and tanX'=-tan(p, 
therefore ^ tanX=/;' tanX', (114.) 
or the tangent of the angle X which the perpendicular arc from the centre of the 
MDCCCLII. 
2 Y 
