274 Mr DAVIES on the Equations of Loci 



whatever point in the equation be taken as the centre of a great circle, that 

 great circle will pass through the pole of the equator *. 



Cor. 6. The intersection of the equator with the circle (1.) will be found 





by combining it with X = ^. This gives 



Cor. 7. If sin X ^ cos p, then cos (6 K) -^ 1, which indicates impossi- 

 bility. 



7T 



Cor. 8. Let, also, ? = H 5 then 



cos(0 K) = O, or 6= ^ + /c (9.) 



the longitude of the point of intersection. 



Cor. 9. All great circles whose centres are in the same meridian, inter- 

 sect the equator in the same points ; for equation (9.) is independent of X, 

 and shews that the point is at the distance of a quadrant from the longitude 

 of the centres in question. It thus agrees with a simple property which en- 

 ters into the elements of the old spherical geometry. 



II. 



To find the distance between two given points on the surface of the 

 sphere. 



Let the co-ordinates of the points be a t ft, and a,, ft,, ; then, if 8 = dis- 

 tance, we have 



cos 8 = cos a, cos a,, + sin a, sin a a cos (ft /t ft,). 



* It is altogether unnecessary to prove, that, for - in these and most of the other re- 

 sults of this paper, we may substitute (2w + 1) ^, where n is any whole number positive 

 or negative. 



