‘al 
503 
geometry of two or of three dimensions) is used as our auai- 
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 ares in the spherical conics. 
*¢ 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 guasi-focal 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 properties are connected 
* Philosophical Magazine for September, 1844. 
