314 Mr DAVIES on the Equations of Loci 



XXIV. 

 SPHERICAL ELLIPSE AND SPHERICAL HYPERBOLA. 



The sum or difference of the arcs drawn from given points or foci 

 on the sphere, to the different points of a sought curve, are given to find 

 the equation of that curve. 



FIG. 18. 



R Q 



Let a, 8 / and a a (3 a be the given points, 2i the said sum or difference, 

 and (f>6 one of the current points of the curve. Then, these two distances 

 are (by II.), 



8, = cos- 1 (cos a, cos (f) + sin a, sin <p cos (Q /3,)} 1 

 8 U = cos- 1 {cos a u cos + sin a u sin <j> cos (6 /3,,)} j 



The condition then becomes 8, 8 a = 2z, and therefore, 



2i = cos" 1 {cos a, cos (p + sin a, sin <p cos 6 /3,} -+- \ 

 H- cos -1 {cos a,, cos + sin a,, sin^> cos /?} } 



Taking cosines of both sides, and reducing the results to their simplest 

 form, we get 



cos* 8, 2cos2i cos 8, cos8 a + cos* 5,, = sin*2i (3.) 



or, restoring the values of and 8 a given by (1.) and reducing the results, 

 we have 



ifaiw f> 



sin 2 2i = cos 2 (p {cos* a, 2 cos 2i cos a, cos a, + cos* a a } + 2cos <f> sin <p x 

 x {sinaXcosa,- 



os a, 2 cos 2 cos a, cos a, + cos a a ] + 2cos < sin < x ^ 



osa, cosa^cosS^cos^ p/+ana,(cora 4 cos a, cos 2i) cos 6 ^} + / ...(4.) 

 ^^cos 2 ^ /3, 2cos2i sina y sina^cos^ /3 t cos6 /3,, + sin* a,, cos 2 6 (3 a } ' 



