Rectification of the Ellipse.^ 45 



►vhence, cos. ' (p= -zr and sin.- 9 =if "tt : it is plain that we must use 



(1_lO (l_i_u 



.1/ "■+ 



.he lower sign, and consequently o is in general less than the half of 



^aken negatively, since cos. (p is negative. 



We have tlien, d^- -2dx(hj cos. 2<p'\'dx^=(fz^ 



and y^—2xy cos. 2;p4"^^ = l I parting y^ —2jy cos, 2vp+x* into 

 its simple factors, and we have 



+«' 



y — .T (cos. 29 — sin. 2^V — l)=e 

 y — x (cos. 29 + sin. 29%/ — 1)=€ "*" 



-z' 



diiTerentiatlng and dy—dx (cos. 2^— sin. 29-/ — l)=*l:<fe'e 



dy — dx (cos. 2$+sin. 2t\/ — 1) =:f<Zz'€'^ 

 multiplying and dy^ —2dxdy cos. 2i> + c?j= = — rf^/^ 



consequently yVr?y ^ —2^j:Jj/ cos. 2l'-|-t^-^^=-'\/— 1=^ 

 and z^=z\/ — I and we have 



y — x (cos. 2?>— sin. 2t\/^^)=€ 



y - .T (cos. 2<J'-[-sin. 2tv/^l)=e"^'^ 



+z' 



-2' 



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

 and then cos.^ 9= _t / — ' and sin. ^9= _i , — — 



and cos.^ m — sm.= 9 = cos. 29= ..1 .— > . 



and we have then dy^ —2dxdy cos. 29+df,r^ =d'^^ 



and y^ — 2^'^cos.294-:r^ = l, by making a6 = -\/iri 



and by the same steps, we have for the hyperbola 



^/:^T^^/^-^ 



y — :r(cos. 29 — sin. 2^ 



y — .r (cos. 29 4- sin. 29 v — lj=e 



by putting a:=cos. 9(:r^+y') 



and - y=sin.o(:r^-yOA/^=l3 it will be seen tliat z comprises 



also f>/dx^ — ^y^j both for the ellipse and hyperbola. 



The equation Ay-+B^'y+Cr-+%+E.r+F=:-0, may take the 

 form y2 +7^2:4-^1^^=0, without losmg its generalit}^ 

 Put ,;2:=a+cos.9(:r'4-y'), 



y=sin.9(:r'-yO; 

 then dx=cos,(p{dxf^dy^)^ 



