414 



And the four corresponding tangential equations are- 



l m n 



+ 7 + — — t; = 0; 



X cos r /a cos r v cos r" 



m n' 



Xcosr fiQQsr fcosr 



V 



m n 



} + ?,= 0; 



X cos r fi cos r v cos r 



V ni n 



+ • : + s=0. 



X cos r fi cos r' ^ cos r" 



(54 

 (55 

 (56 

 (57 



32. The proofs given in Articles 12 and 14 for Dr. Hart's exten- 

 sion of Feuerbach's theorem, it is evident, apply verbatim for the ana- 

 logous theorem concerning circles on the sphere; and the part of i 

 concerning the system of six circles, Art. 12, may also be inferre 

 immediately from the equations (54)-(57) ; for any two of these equa 

 tions are sufficient to determine the ratios X : v and fx : v. Hence th 

 four circles denoted by any pair of the equations (54)-(57) have a com- 

 mon tangential circle, besides the three circles S, S', S". — Q. e. d. 



33. The equations of the inscribed and exscribed circles of a sphe 

 rical triangle may be inferred from equations (4 6) -(49). 



For, denoting the angles at which they intersect 



S' and S" 

 S" and 8 

 S and S' 



by 



it is easy to see that 



2 cos l A 



2 cos \B 



2 cos 



-I 



tan r' tan r" 



m 



tan r" tan r 



tan r tan r 



Hence equation (46) becomes transformed into 



°os kf3- *" cos i B JjL + cos J C J-^-t, = 0. (58) 

 2 A/smr 2 \sinr' 2 A/smr' v 



