BEATTY] AN OUTLINE OF ANALYTICAL SPHERICAL GEOMETRY 2S7 



the circle being defined through the property that two points on it 

 are given. The following analytic method would seem to give us this 

 equation, from first principles. Suppose 

 the equation of the circle through the 

 points P' and P" is required; let P be p 

 any point on this circle; suppose the 

 co-ordinates of these points are in order 

 {6' 4>'), (6" (f>"), and (^ </>). g 



Then 



sin P" P 



s in AP ' 

 sin LP 



sin LP 

 sin AP" 



o 



M N 



That is 



sin P' P sin {6" -6) sin AP' 



cos P' M 



sin P"P sin (6 - d') &m AP" cos P" N 



So too 



Therefore 



sin P" P sin ((/> - (j)') 

 sin P' P sin ((jf - 4>) 



cosP" B 

 cos P' s 



gin {6' - d ) s'm ((f) - (f)') _ cos P' M cos P" R _ cos 4)' cos d" cos 6 coscj ) 

 sin {6 - d') sin {cf)"- </> ) ^ cos P" JV cos P' S ~ cos 6' cos 0" ' cos 6 cos 



Therefore 



Or 



(tan 6" - tan 6 ) 

 (tan (9 - tan d') 



(t an 0^^ - tan ^ ) 

 (tan — tan <f)') 



(tap ^ - tan 6') ^ (tan (^ - tan 00 

 (tan d' - tan ^*) ~ (tan ^' - tan </>*) 



VII. Let us now proceed to find the angular distance between 

 any two points. The simple case where one of these points is the 

 origin will be first dealt with. 



