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3. The Spheric Ellipse. Find the locus of the vertex P of a spherical triangle 
with fixed base FF’, such that the sum of the sides is a constant, p + p’ = 2a. 
Fig. 4. 
This definition defines the Spheric Ellipse MGM'G!. 
Take the origin at the center O of the base FF’. Let FF’ = 2c, p + jp’ 
= 2a,0M = a, OG = Bg. When P falls at G, FG = a = F’G. 
Then from the right triangle FOG (hypotenuse not drawn), we have 
(1) cosa = cos B cose; 
and from PAX, 
(2) tan 7’ = cosé tan 7. 
From the right triangles PAF and PAF’, we have 
(3) cosp = GOSn’ cos (e— —), COSp’ = Ccosn’cos (@ + &). 
Adding equations (3) and using p + p’ = 2a, 
PP! 
(4) cose cos = COS7’ cose cosé. 
5) 
and subtracting (3), 
/ 
PAR: Ue , 
= cOS n sine siné 
(5) sine sin 
9 
f p 
Eliminating ———— and e¢ from (1), (4), (5) and reducing, we find the 
=) 
symmetrical equation of the spheric ellipse 
tan’é tan*7 
= dl = eo 
tan’a tan’p 
a, and 6 being the intercepts on the axes, OM, and OG, respectively. 
Special Cases. (1) Let a = £, and we have a circle 
(A) tan?é + tan’y = tan?a, 
with center at O and radius a. With a = 90°, this circle becomes the bound- 
ary of the hemisphere on which our geometry is located, corresponding to 
the circle with infinite radius in plane geometry. 
(2) Let a = 90°, and the ellipse becomes the two ‘‘parallel lines”, tan?y 
= tan’, passing through the poles of the O Y-axis. 
(3) The equation of a circle upon a sphere may be derived quite readily, 
but the resulting equation is somewhat unsymmetrical. Let &, m, be the 
