Rectification of the Ellipse. 



■nz- 



then (17) - (IS) nz=log. ' 



and(19)-(20) -n^=log.( ^7^1^) ' 



=AG 



1-e 



= -AK 



41 



(19) 

 (20) 



(21) 

 (22) 



consequently, (21) - (22) nz = \og. {±1^ ')=AG+AK = GK 

 equal to the elliptic quadrant. 



Drawing the figure below, and following the description at the 

 commencement, it is evident that we have CD^+2CE CD cos. 

 ACB'+CE2=CF2 or 2/2+2c-c?/ COS. cp+x^ = l. 



Parting the first member of this equation into its simple factors, 



+2; 



we have y-i-x (cos. 9 — sin. (pi/_i) = e and 



—z 



y-{-x (cos. (p + sin. (p\/JZ^)=e+ ; 



differentiating both of these equations, observing from 



xdx sin.^ cp 



dy=- — dx cos. (p^ , :::= =■ that dx and dy have different signs, 



V 1 — a?^ sin.^ (p 



+z 



we have dy — dx (cos. 9— sin. cp'v/ — i)=l<^^e and 



—z 



dy — dx (cos. (p+sin. (py' — 1) ^l^dze'^ 

 multiplying, dy^ —2dxdy cos. (p-\-dx^ = — dz'^ the differential 

 relation of the circular arc AF, and its co-ordinates, as is seen by 



Vol. XVIII.— No. 1. 



