SUPPLEMENTARY CURVES. 181 



p real and q imaginary points, while the same straight line 

 meets the curve B in g' real and/) imaginary points. 



(c.) It is evident from equations (1) and (2) that the 

 diameter 



Ay + Ba7+D = o (3), 



which bisects chords parallel to the axis of y, is common to 

 the curves (1) and (2); hence, if the two conies are supple- 

 mentary in relation to a given straight line, the diameters 

 of the two curves which bisect chords parallel to that line 

 are coincident in direction. 



(d.) Equations (1) and (2) may evidently be written in 

 the form 



(Ay + Bx + Dy=(Bx + 'Dy—A{Cx'+ 2Ea; + F)...(4), 



(Ay + Bx-\-I>f = — {Bx + T>f-^A{Cx''-{-2Ex + F)...(5), 



from which we see (II, 8) that the supplementary conies (1) 

 and (2) touch the two straight lines represented by the 

 equation 



(B" — AC)a;* + 2(BD — AE)ar + D^ — Ar = o (6) 



at the points in which these lines are cut by the common 

 diameter (3). 



When the given curve is a parabola, we have B" — AC = o ; 

 hence, in this case, equation (6) represents only one straight 

 line, which touches each of the supplementary conies at the 

 point in which it is cut by the common diameter (3). 



(e.) From equations (4) and (5) we obtain by addition 



(Ay + Bx + D)» = o,' 



hence (II, c) the curves (4) and (5) have a double contact, the 

 straight line (3) being the chord of contact. Thus we see that 

 two conic sections which are supplementary in relation to a 

 given straight line have a double contact , the chord of contact 

 being the diameter of each curve which bisects chords parallel 

 to the given line. 

 (/.) If equation (5) be denoted by 



