280 



tan 2 £ tan 2 77 



tan 2 a tan 2 /3 



and the spheric asymptoles to either by 



tan £ tan ?; 



tan a tan fi 



5. The Spheric Parabola. A Spheric Parabola may be defined as tin 

 locus of a point moving upon the surface of a sphere so as to he equally distant 

 from a fixed point F and a fixed great circle CM, Fig. 5. 



From the definition PR = PF; let bisect M F. Then from Fig. .">. 



(1) tan 7/ = cos £ tan 77, 



(2) cos PH = sin PR = cos 77 ' sin (c + £), 



(3) cos PF = cos rj' cos (£ — c). 

 Squaring and adding (2), (3) 



1 = cosV (sin 2 (£ + c) + cos 2 (£ — e)}, 

 or 



1 + tanV = 1+4 sine cose sin£ eos£. 

 Substituting from (1), 



tan 2 ?? = 2 sin2c tan£, 

 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 th^ equations in piano by considering the radius of the sphere 

 to increase without limit. This may be done by expressing the arcs in terms of 

 the radius, and finding the limit of the functions in each equation as r — °° . 



For example, in the spheric ellipse, 



tan 2 £ tan 2 ?? 



(1) • + = 1, 



tan 2 a tan 2 /3 



let (£, 77), (a, /3) be radian measure of arcs on a unit sphere; then on a sphera 

 of radius r, we have arcs (x, y), (a, b) determined by 



x y a 



b 



i = -, v = -. a = - /3 



= -. 



r r r 



r 



Elation (1) becomes 





