Rectification of the Ellipse. 43 



adding and log. {x)=~ cos. 9+^1 cos. ^9+^ cos. 3^+ 



+4^ COS. 4?+etc. 



when 9 is a right angle, we have 



, r /-v^/~ , cotan.^ nz cotan.' nz 

 nz=lo^. ^v i^v — (cotan. nz— zj— — -f- ~ 



3^6 



cotan."^ nz 

 Y^ +etc. 



, , . . cotan.' q> cotan*^ (p cotan." m 

 log. (sm. nz)=- 4- _ ^ 



+ 



when tx=y we have 



sin, 2i> sin. 3i) sin. 4^ 

 nz=sm. 9+ "2"+ ~3~+ -^+etc.=AG. 



J , sin. 2^ sin. 3^ sin. 4:p 



and nz=— sm, 9+ -^ — — — 3— +— ^-eic.= — AK, 



, ^ sin. 39 sin. 59 



hence nz=2 (sm. 9+ — ^ — -f ^ +etc.=AK equal to the elliptic 



quadrant ; and tlieir sum is 



^sin. 2s sin. 4? sin. 69 



n5r=2(-2-^+-^+-g^+etc. =AG-AK. 



When 9 is a right angle, w^e have 



.111 

 nz=2\^l ~3+5""7~^tc. equal to tlie circular quadrant 



and AG-AK=0. 



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

 its equations nzA/ — l=Iog. (y+^^ ^ ) and 



nc:v^— I=log.(y+:re '^"') give the co-efficients of 



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

 The equations ytoce ^^ =e ^"^ (23) 



-W- 



and ytoce^ 



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



(24) 



log 



or 



■t 



log. ({tx (sin. 9 a/ ~ 1 ) Iv'l-cc^sin.^?)) ^ ' ' 



