

traced upon the Surface of the Sphere. 289 



XII. 



j 

 To describe a great circle through a given point, which shall make a 



given angle with a given great circle. 



Let the given point N have a / (3, for co-ordinates, and the given circle 

 ML be 



cot^) = tan X, cos (6 /c,) (1.) 



Then also the inclination being given, e,, and the equation of the circle 

 sought, assumed to be 



cot (p = tan X cos (6 K) (2.) 



we shall have (VIII, 6) also the equation 



cos e / = cos X cos \ + sin \ sin X,COS(K /c,) (3.) 



Also, because (2) passes through y /3 /} we have 



cot a,= tanXcos(/3, /c) (4.) 



We are required to find K and X from equations (3) and (4). 

 We may put the equation (4) under either of the forms, 



cos X = sin X tan a. cos (/?. K) 



I (5.) 



sin X = cos X cot a, sec (8, K) J 



Putting these successively in (3), we obtain, after slight reduction, 



cos 6, cos a, 



sin X =. 



cos X, sin a, cos /3, K cos a, sin X COSK, /c 

 cos e. cos a. 



' ' - _ __ ti /g \ 

 (cosX / sin a, cos/3, cos a, sin X, cos/c,) cos/c + (cosX, sin a.sin ft,- cosa, sin X 7 sin K,) sin K 



cosX = _ coee, rina, cos ft-ic _ _ ^ 

 cos X, sin a, cos /3,K cos a, sin X, cos K, K 



cos e, sin a, cos ff, cos K + cos 6, sin a, sin [3, sin K _ 

 ^ina.cosjS, cose^sinX^os/cJcos/c + fcosX^inc^sin/?, cosc^sinX^in/c,) sin/c' 



Butcos*X + sin 2 X = l .................................................. (8.) 



