46 Rectification of the Ellipse. 



sin. (p(dx^ 



writin 



2 ^ _«^2 r>r^^ 2 



/sin.^ (p—m^ COS.-* (p\ 

 and r7y, their values, y^^ -2xy Lin.^ . + m^ cos.~J + 



+ 



(p+ncos.y) . , (2a?yi^ cos, (p+^cos*^) ^./ t 



\ 



sin.2 (p+m^ C0S.2 9 sin.^ 9-f m^ cos*^ 9 



0. 



sm.^' 9+m^ cos.^ 9 

 And dx^ +dy^ :^dz^ =dy''^ +2dx'dy{cos.'' 9— sin.^ (p)+dx'^y the ( 



arbitraries a and 9 authorise us to make 



sin.^ 9 — m^ COS.' (p . ■ 



cos.^ 9 — sm.- 9 



sm.^ 94-wi^ C( 

 and 2am^+n=0. 



1 +w 



From the first wc have cos.^ 9 = —- and sin,^ 9 = t 



Itm 



I f/0 Jl I lit 



and then cos.^ 9- sin.^ 9=^— - = cos. 2cp : 



— n 



and from the second a 



J > 



m^ 



l"^m 



We see from cos, 29= r^;-? by using the lower sign, that in general 



r 



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 y^ —2:r'y cos. 29+0:^2 = 1, (by making 



1 or 



sm.2 9-f ^^ cos.^ (p \7n^ - 1 



and dy'^ —2dx'dy^ cosi2^-^dx^^ =dz^ . 

 From these equations we have, as before, 



y'-a^(cos.29-sin.29V^^)=e^^"S 



y' "a:'(cos. 29+sin. 29'v/^)=e"'^^" ' • 

 When m=0 we have y^ +na:=0, the equation of the parabola, then 



n 



fl = "Q"' cos.2(p = l, sin. 9=0, and consequently the equations above 

 fail in this case. 



The determination of the primitive function f'^dy^ -^dx^, y bebg 

 a function of a?, given by Ay^ -^-Bxy+Cx^ +I)y-{-F,x+F=0, leads 



to the determination of /'/dy'' +dx^j y being a function of a?, given 

 by T{xy)=0'j for parting F{xy) into its simple factors and we have 



{y+cu-\-h){y+cx-{-d){y-\-ex+f) etc.=F(a:y), (A) ; 



». 



