46 Rectification of the Ellipse. 



dy=is\n. ^[dx' — dy') ; we have, after writing for x, y, dx 



/sin.- (p— m^ cos.^ (p\ 



and dy, their values, y'^ -2x'y' { -^ 2^1 ^2 ^^r, ~ +a:'^ + 

 *" ' '^ "^ \ sin. (p+m^ COS. ^ 9/ 



(2fl»i2 c os.(p+wcos.cp) , , (2am'^ cos, (p+ncos.y) ^, , 



sin.^ (p+m^ cos.^ 9 sin.^ cp-\-m^ cos.^ 9 



=0. 



sin.- cp-i-m^ cos.^ tp 

 And dx^-\-dy^=dz^=dy'^-]-2dx'dy'{cos.^ (p—sm.^ (p)-{-dx'^, the 

 arbitraries a and cp authorise us to make 



sin.^ (p—m^ cos.^ 9 



-^. — ;; j ; r— =COS.^ cp — sin.^ 9 



sin.^ 9-}-m2 COS. 2 9 ^ ^ 



and 2am^+n=0. 



1 +m 



From the first we have cos.^ (p=:— — and sin.^ 9 =-3-? 



Urn 

 and then cos.^ 9 - sin.^ 9=^rr~=cos. 29 ; 



— n 

 and from the second a= — r- 



11m . . - . 



We see from cos. 29 = 7-3-' by using the lower sign, that in general 



29 is less than a right angle, and therefore its cosine has the same 



sign as that of 9. When dx and dy have different signs, cos. 9 is 



negative, and we have 3/'^ — 2:c^«/'' cos. 29+x'^ = l, (by making 



m^rt^+wa / m' 



sin.2 9+m2 cos.^ 9"-^ °^' ^~ \m'' -l) )' 

 and dy'^ — 2dx'dy' cos. 2cp-{-dx'^ =dz^ . 

 From these equations we have, as before, 



2/' -a;''(cos. 29 — sin. 29V — l)=e"'^ S 



?/'— a;''(cos. 29+sin. 29^/— l)=e ^ '• 



When m=0 we have y^ ■i-nx=0, the equation of the parabola, then 



— n 

 a=—^i COS. 29= 1, sin. 9=0, and consequently the equations above 



fail in this case. 



The determination of the primitive function f^dy- -\-dx^, y being 



a function of x, given by Ay^ -{-^xy-\-Cx^ 4-D?/4-E«+F=0, leads 



to the determination of /v^dy^ -{-dx^, y being a function of a?, given 



by F{xy)=0; for parting F{xy) into its simple factors and we have 



{y-{.ax-{-b){yi-cx-\-d){y+ex-}-f) etc.=F{xy), (A) ; 



