40 Rectification of the Ellipse, 



^ 



When 9 is a right angle, (5) and (6) become 



COS. nz+ sin. 7iz\/ — i=e 

 COS. nz- sin. wz\/ — i=e 



(9) 



(10) 



and (9) and (10) are sin. 7i 



COS. nz 



I 



2 



(11) 



(12) 



writin 





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



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



(14) 



and y—xe^^ =e ^ ; 



^/=^i=W.rv-^e-^^-0 (15) 



yiTI^log. G-^a^^-') (IG) 



when ^=0 we have y=ll, and 72r=log.(l:l'^"*) ; 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 ^=0 we have a:=tl, and 7icr=log. (:^e''"^'^"0 



e ^"^ ^y ^ commences at A, and terminates at A', com- 

 prising any number of circuits together with AA', 



^og. (+e ^'^ ^y * comprises any number of circuits, commenc- 

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



log* ("f fi^ ^ commences at A and terminates at B' and 



^9v '^v * begins at A, and ends at A', counting nega- 



tively. 



Wh 



(w^iV 



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



writing 



(16) and (IC) and we 



Imve nz=hQ. x{l-e~'^^~')'^'' =AG', (17) 



log.^(l-e*P>^^)^^ = -AG; (IS) 



I 



t 



