StrPPLEMKNTAEY CUEVE8. 



185 



The equation of the conic supplementary to the curve (1) 

 may be found from the last equation by taking »* = o, which 

 gives 

 {A9/ + Bx-\- T>Y = — (Bx + D)*+ A(Caj* + 2Ea; + F)...(4); 



and it is evident (II, c) that the curves (3) and (4) have a 

 double contact, the chord of contact being b — a; = o. 



viir. 



To find the curve which is supplementary to any given 

 curve of the third degree in relation to a straight line parallel 

 to an asymptote. 



If the given straight line be taken as the axis of y, the 

 equation to the given curve will be of the form 



By'x + 2Cyx^ + Dx' 

 + E^' + 2¥xi/ + Gx^ -\- 2}Iy + Kx + Jj = o. 



By solving this equation for y, we obtain 



(1) (Ba; + E)2/ = — (Caj' + Eaj + H) 



±y/{{Cx' + 'Fx + 'H:)^—{Bx+B)(J)x'+Gx' + Kx+L)]y 



and therefore the equation to the required curve is 



(2) (Ba? + E) y = — (Ca;' + Fa; + H) 



+ /{— (Ca;'+Faj-rH)*+(Ba; + E)(Da;' + Oa;'+Ka;+L)}. 



(«) It is evident from the forms of these two equations 

 that the hyperbola 



(Ba; + E)y = — {Cx' + Fx + H) (3) 



bisects every chord of either curve which is parallel to the 

 axis of y. Hence we see that the curvilinear diameter which 

 bisects chords parallel to the axis of y is common to the two 

 supplementary curves, 



[b) By clearing the radicals from equations (I) and (2), 

 we get 



2b 



