42 Rectification of the Ellipse. 



the figure, hence, 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 



wzv/in=log.(y)-^e-?-/^-2pe-^9v^=;^-3^3e-=^9v/-- 

 _— g-4(pv'-i _ etc. and 



-*i^^/^=log. (2/)-"^e<Pv'^-|p==<Pv/^ - ^3e3?v/^_ 



X* .— 



— ^g4(pv/-i — etc. then taking the difference of these two equations 



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



nz=- sin. 9+2^2 si"- 29 + 377 sin. 3:p+^- sin. 49+etc. 

 the sum of these two equations gives 



log- (y) =^cos. 9+2^2 cos. 2^+2^3 COS. 39+^7 cos. 49+ 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 ; the reverse 



X 



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



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

 n2;=lan. nz— ^ — + r — « + etc. 



tan. ^ nz tan.* nz tan." nz 

 and log. (cos. nz)=— ^ + 4 " q "r'^tc. 



when X is greater, we have 



wz\/^=log. (-xe-9\/'^)-^e?%/^ -~-^e^-^V~' - 



^-6^9^/"' — ~-e49v^-._etc.. and 

 3x'3 4:c* 



— war'/— r=log. (— :je9v^-^) — ^-9'/^-;f-;7e--'9^/^ — 

 T /— V^ /— 

 3^-3« 4.^4^ ^^^' 

 the difference of these two equations gives 



»^=log.(te-9v/^)^/^-(| sin. 9 +£:^ sin. 29+ £- sin. 3^4- 

 4-^,1 sin. 494-ctc, 



