SUPPLEMENTAKT CDEVES. 



171 



general lie upon any particular locus; but when the lines 

 succeed one another according to any given law, the repre- 

 sentatives of the imaginary points in which they meet the 

 circle will lie upon a regular curve, which may be considered 

 as the locus of the imaginary points in which the circle is cut 

 by any straight line subject to that law. Now since what has 

 been said of the circle may readily be extended to any given 

 curve, the following problem is naturally suggested : — 



Any plane curve being given, tojind the locits of the imagi- 

 nary points in which it is cut by a series of straight lines 

 drawn in the plane of the curve so as to succeed one another 

 according to any given law. 



In the present state of algebraical science there is little 

 hope of obtaining a solution to this problem, except in parti- 

 cular cases. In this paper I shall confine myself to the two 

 simplest cases, viz., — 1st. The case in which the system of 

 secants is parallel to a given straight line ; — 2nd. That in 

 which all the secants pass through the same point: and I 

 propose to show, by a few examples, that the locus in ques- 

 tion is intimately connected with the given curve in many 

 important properties. 



But before proceeding to the solutions of these problems 

 it may be expedient to establish some general theorems, by 

 means of which the subsequent investigations will be greatly 

 simplified and abridged. 



n. 



'Letu=Ay'+2Bxy+Cx'+2Dy-\'2Ex + \=:o (1) 



be the equation to any conic section, 



V =ay + ^ JJ + 1 = o (2), 



v'=a'y + ^'x+l = o (3), 



the equations of two straight lines ; then the equation 



«i = M + kvv'=o (4), 



where k is an arbitrary constant, will represent a system of 

 conic sections passing througli the four points in which the 



