In-and-circumscribed Triangle. 29 



Richelot has sliown, by precisely similar reasoning, that for 

 circles of the sphere we have 



d^ 



v^cosV— (cosRcos « + sinR sinflcos2A|r)^ 

 d^' 



-v^cos^r — (cosRcos«+ sinRsinacos2'\|r')^ 

 which is of the form 



^l-(X + /i cos 2-^)2" \/l-(X+'/liC0s2'«|r')2' 



where 



cosRcosffl 



A^ 



cos?" 



sin R sin a 



cosr 



And it is very important to remark, that this equation contains 

 the two parameters \, fj,, so that the same equation cannot be 

 obtained with any new values of the parameters a, r ; or the 

 formulte in piano for three or more circles do not apply to cii'cles 

 of the sphere : the geometrical reason for this is as follows, viz. 

 in the plane a circle is a conic passing through two fixed points 

 (the circular points at oo), and consequently any number of 

 circles having a common chord are in fact to be considered as 

 conies, each of which passes through the same four points. But 

 circles of the sphere are not spherical conies passing through two 

 fixed points, but are merely spherical conies having a double con- 

 tact with an imaginary spherical conic (viz. the curve of intersec- 

 tion of the sphere with a sphere radius zero) ; hence circles of 

 the sphere having a common spherical chord are not spherical 

 conies passing through the same four points. I am not sure 

 whether this remark as to the ground of the distinction between 

 the theory of circles in piano and that of circles on the sphere has 

 been explicitly made in any of the treatises on spherical geometry. 



To reduce the equation, write 



tan ^ = A /inM^ tan ^ ; 

 then after a simple reduction, 



Or the relation between the two values of d is 

 d6 d& 



^/ 1- k^ slii2 d " \^\-¥^e'' 



