Rectification of the Ellipse. 43 



y y y \ 



adding and log. (a;)=- cos. <P+2~1 ^^s. 2(p4-o~"H ^'^^- ^f^" 



+4^ COS. 4q3+etc. 



when 9 is a right angle, we have 



. .- /— cotan.^ nz cotan.^ nz 

 wz=log. \y \y — (cotan. nz— o + 5 -^ 



cotan."^ nz 

 ^ +etc. 



cotan. 2 (p cotan.* (p cotan. ^ (p 



log. (sm. nz) = - 2~~+ 4 "~ 6 +^'^" 



when tx^=y we have 



sin. 2(0 sin. 3:p sin. 4ip 

 712;= sm. 9+ —^ — -f — 2 — + ^ +etc.=ALr, 



sin. 2(p sin. 3(p sin. 4^ 

 and wz= — sin. 9-}- — ^ — — — 5 — + — 7 — — etc.= — AK, 



sin. 3^ sin. 69 

 hence nz=2 (sin. 9+ — 5 — + — 5 — -i-etc.= AK equal to the elliptic 



quadrant ; and their sum is 



,sin. 2© sin. 4^ sin. 69 

 n^=2(-2-^+-^+-g-^+etc. =AG-AK. 



When 9 is a right angle, we have 



W2=2(l — o + c — 7~etc. equal to the circular quadrant 



andAG-AK=0. 



With regard to the circle, AB'BA', it is sufficient to observe that 



its equations n^iV — l=log. (y+xe '^'^ ') and 



nz\^ — •l=log.(y+^e^ J give the co-efficients of 



^ y . . 

 the even powers of - or - negative in the developement of nz. 



The equations ytxe ^ =e ' (23) 



and 2/la;e'P^^=e"^^'^ (24) 



include both the circle and ellipse, and we have by eliminating y 



2 

 or z=log.((t« (sin. (p^~~l) tv^l —a:^ sin.29))'^~^ 



