30 On the Purism of the In-and-circumscribed Triangle. 



where 



4/A 



F = 



. tan R s 



,2 _ tan r cos R sin r 



(tanR sin a \^ ,' 

 tan?- cos R sin?*/ 



— 



tan2/3: 



Suppose that for ■<^ = 0, '<|^' = /S, it is easy to see that 

 sin^ (R — «) — sin^r 



cos^ R sin^ r 



Let the coi'responding value of 6' be 6' = ct, i. e. suppose that 

 for ^=0, we have 6' = u, then 



^ cos (R + fl) 

 2 _ cos r sin^ (R — a) — sin^ r 



, cos(R— a) ' cos^Rsin^r 

 cosr 

 _cos?* — cos(R + «r) cos^?-— cos^ (R — «) 

 cos r — cos (R — «) cos^ R sin^ r 



_ (^cosr — cos (R + fl))(^cos?-+ cos (R — «)) 



cos^ R sin^ r 

 _ (cos r + sin R sin «)^— cos^ R cos^ a 



cos'^ R sin^ r 

 _ (cos r sin R+ sin a)"^— cos^ R sin^r 



cos'"^ R sin^ r 



wh 



ence 



^ „ /tanR sin « \^ , 



tan2«=(- + ^^--— ) -1, 



\ tan r cos R sm rj 



/tan R sin a \ 



sec « —\ 1 n — : — ) • 



\ tan r cos R sni r/ 



And a having this value, the finite relation between 6, 6' is 



By comparing with the corresponding formula in piano, we 



arrive at Richelot's conclusion, that the formulaj for the sj)here 



may be deduced from those in piano by writing in the place of 



R a ^, ., ^. tanR sin« . , 



— , — , the functions , ^^— ^ — ■-, respectively. 



r r tan r cos R sni r ^ ^ 



2 Stone Buildinj^s, 

 October 1, 185(i. 



