280 
tan? tan? 7 
ee ae 
tan? a tan? 8 
and the spherie asymptoles to either by 
tan é tan 7 
EES as SAS 
tan @ tan p 
5. The Spheric Parabola. A Spheric Parabola may be defined as the 
locus of a point moving upon the surface of a sphere so as to be equally distant 
from a fixed point F and a fixed great circle CM, Fig. 5. 
From the definition PR = PF; let O bisect M F. Then from Fig. 5, 
(1) tan 7’ = cos é tan 7, 
(2) cos PH = sin PR = cos 7’ sin (e + &), 
(3) cos PF = cos 7’ cos (E — @). 
Squaring and adding (2), (3) 
1 = cosy’ \sin? (£ + ¢@) + eos? (E — @)), 
or 
1 + tan?n’ = 1 + 4 sine cose sin€é cose. 
Substituting from (1), 
tan?y = 2 sin2e tané, 
which is the required equation. 
6. Correspondence to Plane Geometry. The above equations of the 
spheric straight line, ellipse, hyperbola, parabola, and circle, show a marked 
similarity to the corresponding equations in the plane. These equations may 
be reduced to tk» equations in plano by considering the radius of the sphere 
to increase without limit. This may be done by expressing the ares in terms of 
the radius, and finding the limit of the functions in each equation asr “= %. 
For example, in the spheric ellipse, 
tan?é tan?7 
qd) —— + — =1, 
tan*a tans 
let (£, 7), (a, 8) be radian measure of arcs on a unit sphere; then on a sphere 
of radius 7, we have ares (x, y), (a, b) determined by 
x y a b 
E == 7) =, 2. => =; 5S = = 
r r r r 
Eyuation (1) becomes 
