Rectification of the Ellipse. 45 



whence, cos.'^ 9=— rr and sin.^ 9 =+"37 : it is plain that we must use 



CtiO CI 1,0 



the lower sign, and consequently a is in general less than the half of 

 a right angle, and therefore cos.^ 9 — sin.^ (p = cos. 2?i = ~:7 is to be 



taken negatively, since cos. 9 is negative. 



We have then, dy '■ — 2dxdy cos. 2(p-\-dx^ =dz^ 

 and y^—2xy cos. 2cp-{-x^ = l -, parting y^ —2xy cos. 29+^^ into 

 its simple factors, and we have 



I/ — a: (cos. 29 — sin. 29V —l)=e 



-zi 



y — a: (cos. 29 + sin. 29\/ — l)=e + 



differentiating and dy—dx (cos. 29 — sin. 2(jpV —l)=tdz'e 



dy — dx (cos. 2^+sin. 2^V — l)=ldz'e^ 

 multiplying and dj^ —2dxdy cos. 2^-\-dx^ = — dz'^ 

 consec[uent\y f\^ dy^ —2dxdy cos. 2f-\-dx^=z'\/^^i=z 

 and z'=z\/^—i and we have 



y — x (cos. 2?— sin. 2f\/'^^)=e 



y-x (cos. 2'J' + sin. 2$\/_i) = e ^ 



When h^ is negative, we have h^x'^ —a"y^=a-b^, 



a . -b\/ — 1 



and then cos.^ 9= _7 / — = and sin.^ 9— _/ , — =- 



^ a^bv —1 a+o\/ — 1 



ath\/ — 1 



and COS.- 9 — sin.^ 9 := cos. 2(p=~ _7 . — ? 



(i|^^0^/ — 1 



and we have then dy"^ —2dxdy cos. 2'^-\-dx^ =dz^ 



and y^ — 2ry cos. 2ip-\-x^ =1, by making ab = \/ — 1, 



and by the same steps, we have for the hyperbola 



y — x(cos.29 — sin.29V — 1) =e ' 



?/ — .r(cos. 29 + sin. 29V — l)=e ^ 'j 

 by putting a;=cos. 9(5/+?/') 



and 3/=sin. 9(x' — 3/')'(/"ZrT, it will be seen that z comprises 



also fvd.c^ ~~f^yU both for the ellipse and hyperbola. 



The equation A3/"+B.ry+C.r-+Dy-f Er+F=0, may take the 

 form ?/2 -{-nx-^-'mx^ =0, without losing its generality. 



Put :r=G+cos.9(:r'+2/0' 



y^sm.(p{x'-y'); 

 then dx— COS. (D{dx'-\-dy'), 



