42 Rectification of the Ellipse, 



the figure, lience, what has been said of the ellipse, AB'BA^ is equal- 

 ly applicable to the circle AB'BA^ 



Developing (15) and (16,) y being treated as the greater, we have 



^^ I 



g-4 (pv- 1 _ etc, and 



4y^ 



X 



i 



^/~l = \og.{y)-y^^-^-|^^^^^ 



Ay 



^g4<pV'-i_ ^tc. then taking the difference of tliese two equations 

 and observing the equations (11) and (12), we get 



nz=- sin. 9+2I2 s"^"- ^?+3^"? ^^^* ^^^Ay^ ^^^- 4? + etc. 

 the sum of these two equations gives 



X x^ x^ x^ 



^o§- (y) =y^^^-^+2^2 ^^^' 29 + 377 cos. 3?+— 7 cos. 4?+ etc. 



When 9 is a right angle, the sines of the even multiples of 9 are 

 equal to zero, and of the odd, alternately plus and minus j the reverse 



X 



happens to the cosines ; the ratio - takes a name, and then 



tan.^ nz tan.^ nz tan '^ nz 



72;r=tan» nz— ^ — +"5" " — "7 + ^^^• 



_ _ , . tan.^ nz tan.^ nz tan.« nz 

 and log. (cos. nz)=- — ^ + — 4 — - — g +etc. 



when X is greater, w^e have 



nz 



\/-l=log. ("^:re-?A/-')~-e9v^'' - J^-e^?-/"" 



^ ^ ^ X 2x'^ 



i 



the differerice of these two equations gives 

 nz=los.(te-7v/-)v/^-(| sin. <p+£-, sin. 2^+ £- sin. 3?+ 



+4;^'4 sin.4?-f-etc. 



