the Elliptic Equation. 



201 



, . , sin a /111. 



Returning therefore to the first case m which -.— g = smt/,let 



us project the circle from the south pole stereographically on the 

 plane of the equator. Let m be the distance between the cen- 

 tres of the sphere and of the stereographic circle, and n the 

 radius of the stereographic or sabcontrary circle. We have at 



once 



si nat+sm^ 



sin a— sin /3 

 m-/i=tanl(«-^)=^^^^^^^^^, 



m = 



sin a 



sm, 



whence 



cos « + cos yS ' 



cos « + cos /S' 



m 



— = sm 



/I 



The centre of the sphere is therefore the centre of similitude 

 to the family of circles subcontrary to those on the sphere whose 

 modulus is sin 6. Indeed, it is evident that the axis of the 

 sphere is an axis of similitude to the family of stereographic 

 cones having the family of circles for their bases. 



Let 2<r be the ^\^ of the subcontrary circle intercepted by 

 ' radius 



the meridian planes of longitude zero and ^. In the annexed 

 figure, is the centre of the circle, C of the sphere, M C P is 

 the longitude, M0P = 2'>/r. We have 



sin 2-^ 



tan 



PM n sin 2-\/r 



^~ MC~ncos2-«/r + m'~cos2'»/r+sin^" 



M O C 



If we change the variable from (/> to ->|r in the expression 

 </0 



(1 - sin'-' e sin^ <^)^ 

 d(f> 



-, we have 



(1 



- sin^ sin« <}>y^ 1 + sin ^ I J 



dyfr 

 4 sin 6 



'^y 



