272 Mr DAVIES on the Equations of Loci 



the other, on account of a closer alliance to the form in which the data is ex- 

 hibited. Whether it may be generally so or not, future experience alone can 

 decide ; but my own observations lead me to think, that the polar method is 

 capable of greater facili ty of application to spherical loci than the geographi- 

 cal. But, as the radius-vector and polar angle are so related to the latitude 

 and longitude, we may at any step change from one to the other considera- 

 tion, without the slightest mental effort, and without any analytical reduc- 

 tion whatever. 



I. 



THE EQUATIONS OF THE CIRCLE ON THE SPHERE. 



To find the equation of a circle whose centre and radius on the surface 



of a sphere are given. 



Let M be the centre, P the pole, EQ the equator, PE the first meri- 

 dian, and L any point in the circumference, whose equation is sought. De- 

 note by X and K, the radius-vector PM and polar angle EPM of the centre 

 M, and by g the radius ML of the circle. Let EPL = 0, and PL = <t>. 

 Then we have at once 



cos fucos A cos 0.+ sin X sin (f) cos(0 K), ...(!.)* 



which is the equation required in its most general form. 



* See, for other forms, Note A, at the end of this paper. 



