SUPPLEMENTARY CURVES. 



175 



and by eliminating a;^ and y^ from these three equations we 

 shall have 



A,E/+C,D,'+F,B;-2B,D.E— A.C.F, = o (8), 



which is an equation of the third degree with respect to 1c, 

 since A^, B^, C^, &c., are linear functions of k. Now since 

 every equation of the third degree has at least one real root, 

 it follows that one of the values of k deduced from equation 

 (8) must be real ; and if this value be substituted in equations 

 (6) we shall obtain the corresponding values of x^ and y^, 

 which will also be real, since equations (6) are of the first 

 degree. Hence we see that any two conic sections (1) and 

 (2) described in the same plane have at least one system of 

 conjugate common secants, the point of intersection of which 

 is necessarily real, although the secants themselves may be 

 imaginary. 



V. 



If a straight line AP be cut by a given curve of the wth 



degree in the points Q^, Q^, Q^ Q„, it is required to 



express the ratio 



PQ, . PQ, . PQ3 PQ„ : AQ, . AQ, . AQ^ AQ«, 



which is compounded of the ratios of the segments of A P, 

 in terms of the co-ordinates of P referred to any rectangular 

 axes passing through A. 



Let x', y be the co-ordinates of P, and put A P = r, 



Ad, = p,, AQ^ = p,, AQ3 = P3 ACl„ = p«; then, if 



we adopt the notation 



(PQ„) = PQ, . PQ, . PQ3 P<^«> 



we shall have ff-^ ^ (^-P>) (^-pj (^-pJ. (^-P») , 



(AQ„) Pi ' P^ ' Pz P» 



and when n is even this reduces to 



ff.c^l=,_(±+_L+4,).+(-LH.J-+-i-+fc>.. 



(AQ„) ^Pi P^ ^ VxP, PiPb p^Ps ' 



— &c (1). 



Let w = o be the equation of the given curve ; then 



