316 Mr DAVIES on the Equations of Loci 



sect that distance. Then, we shall have a, = a a = 5 ; and /?, + /?, 

 (= 7. suppose). Hence, 



sin a t = sin a,, = 1 

 cos a, = cos a a = 

 sin /3, = sin 7 

 sin/3 // = + sin7 



COS /3, = COS /? =r COS 7. 



These reduce (3.) to 



sin d) = -+- 



sin ^ cos i 



V cos 2 7 sin 2 i cos* 6 + sin 2 7 cos 2 i sin* 0'' 

 Suppose 7 0, or the foci, to coalesce : then 



cosi 



.(6.) 



sin ^ cos i 

 sin i cos i 



I * Dill 



sm0 = -_=-,-- 



"" - , 6 cos 6 



.(7.) 



the equation of a less circle, whose radius is i, whose centre is at the point 



7T 



\ = - and /c i= 0. 

 x 



The general equation becomes still more convenient than in (5.), if we 

 take the pole midway between the foci, and the prime meridian at right 



FIG. 19. 



angles to the circle joining the foci. For, then, we have a, = a,, (= a sup- 

 pose), and /?, = J, & - + |. Then, 



cos 6 ft, = cos 6 + g =: sin 6, and cos 6 /3 a sin 6. 



