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V. On the Foci and Directrices of the Line of the Second 
Order. By T.S. Davies, F.R.S. Lond. and Ed., F.S.A., 
§c., Royal Military Academy. 
N vol. viii. (pp. 317-21.) of Gergonne’s Annales des Mathé- 
matiques, M. Bret has proposed a method of discovering 
the foci of a conic section, defined by the usual general equa- 
tion of the second degree between x and y. He adopts as his 
definition of the focus, that of Euler: the focus is a point in 
the plane of the curve, such that its distance from any point 
in the curve is a linear function of the corresponding coordi- 
nates of the curve. The investigation, however, is not carried 
beyond rectangular coordinates, and he does not solve his 
resulting equations except for two very simple cases. ‘Those 
equations are certainly, to the eye, very simple; but on at- 
tempting their solution, it will be found that they involve an 
amount of reduction such as we should be little prepared to 
expect. Moreover, the conclusion he draws from his equa- 
tion (that there are four foci, two real and two imaginary) 
does not seem to be justified by the investigation about to be 
here offered. 
Sir John Lubbock has also investigated, under a view which 
is virtually the same as Kuler’s, but by a very elegant process 
peculiar to himself, the existence of a focus in the line of the 
second order*. As the subject is introduced incidentally and 
briefly in Sir John’s paper, there is nothing to lead us to any 
conclusion as to how far he had pushed these researches. 
These two are all the discussions of the problem of the 
foci with which I am acquainted, though possibly others might 
be scattered in some of the continental periodicals which I 
have no means of consulting. As the method which I have 
employed is perfectly general, and I have completely resolved 
the resulting equations, giving at the same time both the foci 
and directrices, instead of the foci alone, I am disposed to 
think it will not be without interest to those geometers who 
have been in the habit of studying the conic sections by means 
of their coordinate equations, One or two applications of 
the formul here obtained may be given in a future Magazine, 
THEOREM. 
A point and a line can be found in the plane of a line of the 
second order, referred to any system of rectilinear coordinates, 
such that any point in the line of the second order shall have its 
distances from the point and line to be found in a constant 
ratio, which can also be found. 
* Phil. Mag. S, 3. vol. x. Aug. 183], p. 86. 
