SUPPLEMENTARY CURVES. 



191 



Let p and e denote the polar co-ordinates of the point x^ y; 

 then, by dividing the preceding equation by p*, we obtain 



Asin''5+2Bsin^cos^+Ccos»^+2(Dsiad+Ecos^)p + F{p)''=o; 



. • . F ; p = — (D sin 5 + E cos d) 



±/{(Dsin^Ecos<9)*— F(Asin*^+2Bsin5cos5+Ccos»^)}..(l) 



is the polar equation of the given curve, and by changing the 

 algebraic sign of the quantity under the radical sign, we 

 shall have 



F : p = — (D sin 5 4- E cos ^) 

 ±v/{F(Asin'^+2Bsin^cos5+Ccos*^}— (Dsin^+Ecos^)'}..(2) 



which is the polar equation of the required locus. 



{a) Since any value of 6 which gives real values of p in 

 equation (2) will give imaginary values of p in equation (1), 

 it follows that the given curve (1) may be considered as the 

 locus of the imaginary points in which the curve (2) is cut by 

 any system of straight lines passing through the given point. 

 Hence if any straight line passing through A cut either of 

 the curves in two real points, these may also be considered as 

 imaginary points on the other curve. From the analogy of 

 this property to the one obtained above (VI, a), the curves 

 (1) and (2) may be considered as supplementary in relation to 

 the given point A. And, in general, any two curves may he 

 said to he supplementary in relation to a given pointy when 

 any straight line drawn through the point meets one of the 

 curves in p real and q imaginary points, which can also be 

 considered as p imaginary and q real points on the other 

 curve. 



{h) By passing from polar to rectangular co-ordinates, and 

 by clearing the radicals, equations (1) and (2) become 



(Dy + Ex+F)*=— F(A3r'+2Ba:2^+Car»)+(D3^+Ear)\..(3), 



(D^ + Ea: + Fr=F(A2^'-l-2Ba:y + Ca,■•)-(Dy+E.r)^..(4), 



