170 



MR. ROBERT FINLAY ON 



theory exclusively, but my object is not so much to defend 

 this theory as to illustrate and extend it. 



Section I. — Preliminary Propositions. 

 I. 



If the circle a;' + ?/* = a* be cut by a straight line a? = c, 

 the ordinates of the points of intersection will evidently be 



+ \/(a^—c') and — \/(a'~c'), 

 which are both imaginary when c is greater than the radius a. 

 Thus we see that any straight line in the plane of the circle, 

 and at a distance from its centre greater than the radius, cuts 

 the circle in two imaginary points. Now it has been proposed 

 by M. Poncelet, (Traite des Proprietes Projectives des Figures, 

 p. 29.) that the real points 



X = c, y =^ V^(c' — «0 and x =^ c, y = — l/(c* — a*), 

 obtained by changing the sign of the quantity under the 

 radical in the imaginary expressions given above, should be 

 taken as the representatives of the imaginary points in ques- 

 tion. Again, since 2 y/^O'"' — c") may be considered as the 

 imaginary chord which the circle intercepts on the given 

 line, M. Poncelet has proposed that the real expression 

 2 V^(c* — a*) should be considered as the ideal chord inter- 

 cepted by the circle x' + y'' z=z a"" on the straight line a? = c. 

 He has also applied the term ideal chord to the indefinite 

 straight line x = c, when its intersections with the circle are 

 imaginary : hence any straight line in the plane of a circle 

 may be considered as the chord of the circle ; but, for the 

 sake of distinction, it is called a real chord when it cuts 

 the circle in two real points, and an ideal chord when its 

 intersections with the circle are imaginary, or when it lies 

 wholly without the circle. 



These principles being admitted, if an arbitrary series of 

 straight lines be drawn, each of which cuts the circle in two 

 imaginary points, the corresponding real points will not in 



