Rectification of the Ellipse. 39 



Consequently, 2/— a?(cos. (p — sin. (pv—l)=e '^ S (1) 



and ?/— a;(cos. (p+sin. <p v/— l)=e '^ '? (2) 



also, accenting, ?/'— a;'(cos. 9 — sin.9V — l)=e^ ^ '» (3) 



and ?/' — a:;'(cos. cp+sin. 9 V — l)=e ^ '? (4) 



then (1), (3), 2/j/'+a?a:'(cos.29 — 2cos. 9sin.(pv'3ri — sin.^cp) 



— (i/'a;+?/a:')(cos.9 — sin.9\/_ i)=:e '' 



and (1), (4), yy'-{-xx' — [y'x-\-yx')cos.(p — {y'x — yx') ^m.(^\/'ZZ\ 



=e • Put a and ^ for the co-ordinates of the arc {z-\-z') 



and a' and /S' for those of [z — z'), 



and then /3 — a(cos. 9 — sin. 9V — l)=e "^ 



and jS'' — Dt'(cos. 9 — sin. 9'^ — l)=e • Consequently 



yy' -\- a;a?'(cos.-9 — 2 cos. 9 sin. (^^/ZI\ — sin. ^9) — [y'x + ya:') (cos. 9 



— sin. 9\/IIl) =/3-a(cos. 9 — sin. 9\/ HT) and yy'-\-xx' — {y'x-\-yx') 

 cos. 9 — [yx' —y'x)^m.. 9^/ _ i=/3' — a'(cos. 9 — sin. 9\/ — 1). 

 Comparing the homologous parts of these equations we have 



a=y'x-\-yx' — 2xx' cos. 9, 

 ^=yy'-xx'', 

 a'=:y'x — yx', 

 (3' =yy' -]-xx' - 2yx' COS. q}. 

 When 9 is a right angle, cos. 9=0, and the ellipse becomes a circle, 

 and the co-ordinates take the name of sine and cosine : we then have 

 sin. (2'-f-2;')=cos. z' sin. sJ+cos. zsin. 2;'; ' 

 cos. [z-{-z')=cos. z COS. 2;'— sin. 2; sin. 2r' ; 

 sin. [z — z')=cos. z' sin. 2r — cos. zsin.2;' ; 

 COS. {z - z''):=cos. z COS. ^'-j-siu. zsin. z'. 

 Writing nz for z, in (1) and (2) we have 



?/— a; (cos. 9 — sin. 9'v/-i^) = e and (5) 



- z /Z — 



y—x (cos. 9+sin. (p^-~i) = e ; (6) 



X and y being the co-ordinates of any multiple, nz of the arc z, 



e — e /«,\ 



from which (6) — (5 ) x~ -^ — = 7^== — *. ' J 



^ J y J 2 sm. 9\/ — 1 



and (6) + (5) 4-^ cos. 9 



,._ e"^^~'+e~"^ +cotan. 9v/— l(e"^'^"'-e""^^ "') 



