184 



MR. EOBEBT FINLAY ON 



one or both of the conies may vary with any variation in the 

 fundamental data. This consideration may perhaps remove 

 Mr. Salmon's objection to this theory, which he has founded 

 on the fact that any conic section has an infinite number of 

 supplementary conies. 



{k.) If a system of conic sections have an ideal common 

 secant, it will be a real common secant of the system of 

 conies which is supplementary to the given one in relation 

 to any straight line parallel to the common secant. For, 

 since every curve of the given system cuts the ideal chord 

 in the same two imaginary points, these will evidently be 

 real points on every curve of the supplementary system. 



VII. 



If two conic sections have a double contact, and if their 

 supplementary conies be taken in relation to the chord of 

 contact, the supplementary conies will have a double contact 

 at the same points as the first two conies; so that, if the 

 chord of contact be real for the one system it will be ideal 

 for the other. 



Let the axis of y be parallel to the chord of contact, then 

 if the equation to one of the conies be 



Ay^ + 2Bxy + Ca:^ + 2Dy + 2Ex + F= o (1), 



the equation of the other conic will be (II, c) 



Ay^ + 2Ba;y+ Cx^ + 2Dy + 2'Ex + ¥ =^m{b — x)\..{2), 



where b — x = o denotes the chord of contact. Now from 

 equation {2) we readily obtain 



Ay=: — {Bx + I>) 



±^{{Bx + By — A{Cx^ + 2Ex + F) + Amib^-xY}, 



from which it is evident (VI, a) that the equation of the conic 

 supplementary to the curve (2) will be 

 iAy + Bx + D> = —{Bx + D)^ + A (C^'- + 2Ea;+ F) 

 — Am{b — xY (3). 



