278 



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 MGAI'd 1 . 

 Take the origin at the center of the base FV. Let FF' = 2c, p + p' 

 = 2a, OM = a, OG = 0. When P falls at G, FG = a = F'G. 



Then from the right triangle FOG (hypotenuse not drawn), we have 

 G) cos« = cos /3 cose; 



and from PAX, 

 (2) tan r)' = cosf tan -q. 

 From the right triangles PAF and PAF', we have 



(3) COSp = cos?/ cos (c — £), cosp' = costj'cos (c + £). 

 Adding equations (3) and using p + p = 2a, 



p — p 



(4) COSa COS = COS77' cose cos£. 



and subtracting (3), 



. p ~ p , . 



(5) sma sin = cos -q sine sin£ 



p — p 

 Eliminating and c from (1), (4), (5) and reducing, Ave find the 



symmetrical equation of the spheric ellipse 



tan 2 £ tan 2 77 



+ = 1, 



tan 2 a tan 2 /3 



a, and being the intercepts on the axes, OM, and OG, respectively. 

 Special Cases. (1) Let a = (i, and we have a circle 

 (A) tan 2 £ + tan 2 i7 = tan 2 «, 

 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 li?ies", tan 2 ?; 

 = tan 2 /?, passing through the poles of the OY-axis. 



(3) The equation of a circle upon a sphere may be derived quite readily, 

 but the resulting equation is somewhat unsymmetrical. Let £1, 771, be the 



