344 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTTC INTEGRALS. 
spherical conic, on the arc of a great circle touching it, makes with the principal 
major arc, is inversely as the radius of the sphere. 
A simple geometrical construction will give the magnitude Fig. lo. 
of those angles 1. and A'. Let the ellipse OAB be the base of 
the cylinder; OCC',ODD' being the bases of the hemispheres 
whose intersections with the cylinders give the spherical 
conic sections with reciprocal parameters. Erect the equal 
tangents DP, CQ, and join PO, QO. The angles AOP, AOQ ^ 
are X and X'. 
When DP=CQ=0, X=:X'=:0; when DP=CQ=oo, 
X=l!=-. The condition (109.) shows that when k=a, ® 
k'=oo . Now as k' tanX'=a tanX, is finite always, so long as X is not absolutely 
in order that its equal k' tanX' may be finite also, we must have X' always equal to 0, 
for every finite value of tanX. 
XXXIII. The tangent of the principal arc of a spherical parabola is a mean pro- 
portional between the tangents of the principal arcs of two spherical conics with 
reciprocal parameters ; the three curves being the sections of the same elliptic 
cylinder by three concentric spheres. 
Since tan"a=^^ 23 ^, tanV^p^g, tanV tana'= 
Introducing the equation of condition (109.), we get 
tana tana'=-, 
(115.) 
Let A:" be the radius of the sphere whose intersection with the cylinder gives the 
spherical parabola; then k"‘^=a^{\-\-i). See (96.) 
Plence 
aH-, and tanV'=p 2 ^^=^ ; therefore 
tana tana' = tan V' (HO-) 
The altitudes of the vertices of the three principal Fig- H- 
major arcs of the two spherical conics with recipro- 
cal parameters, and of the spherical parabola, above 
the plane of the elliptic base of the cylinder, are in 
geometrical progression. Let AQ be the altitude 
of the vertex of the major arc of the spherical para- 
bola. AP, AR the corresponding altitudes of the 
vertices of the major arcs of the spherical ellipses. 
Then AP=\/F— a^, AR^^/c'^— a®, AQ=\/k"'^—a^=a>syi. The equation of con- 
dition gives, as in (109.), APx AR=AQ^^. 
We shall give, further on, an expression for the sum of the arcs of two spherical 
conic sections having the same amplitude, but reciprocal parameters. 
