503 



geometry of two or of three dimensions) is used as our auxi- 

 liary. 



" Beginning with curves of the second degree, we may 

 assume the properties of the circle as known. Then, 



"1. If we operate on the circle by the cyclo-polar method, 

 we arrive at the modular generation of the conic sections, and 

 are enabled to discover and prove almost all the properties re- 

 lating to their foci and directrices. This mode of proceeding 

 has been explained and practised by Poncelet, Gergonne, and 

 others, and is familiar to all students of geometry. 



" 2. Let us next place the centre of our auxiliary circle at 

 the centre of the conic, and perform a second transformation ; 

 and we shall thus deduce from the already proved focal pro- 

 perties a series of theorems relating to two remarkable lines, 

 which may be termed the secondary directrices of the conic. 

 These lines have already been the subject of some researches 

 of Dr. Booth ; but he has obscured the simplicity of their 

 theory, by presenting their properties as results of a peculiar 

 and somewhat intricate analytic method. I have already no- 

 ticed, in another place,* the striking analogy which exists be- 

 tween these lines and the cyclic arcs in the spherical conies. 



" 3. But, instead of placing the origin of transformation 

 at the centre of the conic, we may suppose it situated anywhere 

 in the same plane ; and in this way we shall be led to consider 

 a conic section as the locus of a point which moves so that the 

 square of its distance from a given point is constantly propor- 

 tional to the rectangle under its distances from two fixed right 

 lines. The transformation at the same time indicates a class of 

 new properties, which might be called quasi-{oc&\. properties, 

 inasmuch as in a particular case they reduce to the ordinary 

 focal properties, and they are in all cases analogous to those of 

 umbilicar foci in surfaces of the second degree. The generation 

 of the conic sections with which these propertiies are connected 



* Philosophical Magazine for September, 1844. 



