traced upon the Surface of the Sphere. 



281 



VII. 



To find the angle made by a given great circle witk a given meridian. 



Let the given circle and the given meridian be M F and M D, and de- 

 note them by the equations 



cot(p = tan X cos (6 K) (1.) 



= (2.) 



When they intersect, we have 



cot< = tan X cos (/3, K) (3.) 



Again, when the given great circle intersects the equator, we have 

 cos ( 6 K) = 0, or 6 = + K = F D, or 



FD=|+/c-/3, .'.(4.) 



But, by Napier, sin M F tan F D cot F M D, or putting the angle = e /5 

 we have 



_ sin MF _ cos (f> 

 ~tanFD~ cot (*-/?,) 



_ cos cot- 1 ( tan \ cos ft, K) 

 cot(/c ft)~ 



, tan \ sin K (3, ^ 



~Vl+tan z Xcos*/c /3, 







sin K p, sin X 



