RecUf cation of the Ellipse. 39 



Consequently, y— a:(cos. ? — sin. (pV^— i)=e '^"S (] 



and y— ^(cos.<p + sm.9 v^— l)=e '^ ' ? (2) 



also, accenting, y'— ^'(cos.? — sin.ipv — 1) — e ^"S (3) 



and y-a?^(cos. (p4-sin.<p V - 1) — g " v « j /^\ 



+ 



9\/ — 1 -sm.^9) 





+a?a?^— (^0?+ 2/^0^0^' ? ~ ilf^ " y^O sin,(pv/ 



1 



e ^ '. Put a and ^ for the co-ordinates of the arc (z-f z') 



and a' and /3' for those of (z — r 



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



and /3^ — a'(cos. cp — sin. 9 v ^ 1) =e v " ' . Consequently 



' 4- ^J^(cos. = 9— 2 COS. 9 sin. <^\/ZI\ — sin*29) — (y'^ + y-^/) (cos. 9 

 sin. 9>v/iri) =/3-a(cos. 9 — sin. 9^/ HT) and yy^-^-xx' — (/^+y^) 

 COS. 9 - [yx^ —y^x)sin, 9 v/ — 1 =/S' — a'(cos. 9 - sin. 9^/ 

 Comparing tlie homologous parts of these equations ^ye have 



a=y^x-{-yx^ -^^xx^ COS. (pi 



^=yy^ — xx' \ 



a^=y^x — yx^y 



l3^=yy^-\'Xx'- 2yx^cos.cp. 

 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. (2r-f'2^') = cos. z^ sin. z-j-cos. zsin.z^^ 



cos. (r-f z') = cos. z cos. z^ — sin. z sin. z^ ; 



sin. (z — s') = cos. z^ sin. z — cos. zsin. z' ; 



Wr 



cos. (z - z^— COS. z COS. z'+sin. zsin. z\ 



y ^x (cos. 9 — sin. 9\/-i) = e and (5) 



w— 0? (cos. 94-sin. 9v^^) = e j (6) 



a? and y being the co-ordinates of any multiple, 7iz of the arc z. 



,.zv/-._^--v/-. 



from which (6) — (5) x~' — 7^ — = 7^^^= — 



\ ^ \ / 2 sm. 9v/ — 1 



and (6)4-(5)+'3? cos. 9 



gnzv »_^g ^ -t-cotan.9.v/ — 1 (c ^ — e 



(7) 



(8) 



