40 Rectification of the Ellipse. 



When (p is a right angle, (5) and (6) become 



COS. nz-\- sm. nz-\/ -^i = e (9) 



COS. nz- sin. W2'v/Zri=e "^"^ ^ (10) 



and (9) and (10) are sin. nz= _- (11) 



COS. nz= — (12) 



writing 1 for n and (p for z, then (9) and (10) become 



COS. (p+sin. (p\/1Ii=e 



and cos. (p— sin. 9 =e ^ ' ; we may write therefore 



(5) and (6) thus: y-xe''^^'' =e'''' ^'' (13) 



and y-a;e*^'^^=e~"^^^; _ ' (14) 



hence, nz\/ zri=\og.(^y — xe~^'^ ~^) (15) 



— «z\/zri=log. {y—xe^'^'^) (16) 



whena;=0 we have 3/=tl, and w2;=log.(ll ) ; the upper sign 



indicating any complete number of circumferences, and the lower 

 sign any odd number of semi-circumferences, as is easily seen by 

 counting from A positively. 



When y=0 we have a?=ll, and w2:=log. (:fe+'^'^~')'^~' ; 



log. ( - e ^ ' ) ' commences at A, and terminates at A', com- 

 prising any number of circuits together with AA% 



log. (+e ^ ') * comprises any number of circuits, commenc- 

 ing at A and terminating at B', counting positively, 



log. (4 e ') ' commences at A and terminates at B' and 



log. ( — e^^ ') begins at A, and ends at A', counting nega- 



tively. 



When (p is a right angle w2^=log. \:^\/^) 



When tx=y it indicates the vertex of the greater and lesser axis, 

 as is seen by making the first member of y^ -\-2xy cos. 9+a?^ =^CF^ 

 maximum; writing therefore la; for y, in (15) and (16) and we 



havenz=log. a:(l-e"''''^'')^"'=AG; (17) 



-w^=log. a;(l-e^^'^)^~ = -AG; - (18) 



