393] 
51 
393. 
ON THE CONICS WHICH TOUCH THREE GIVEN LINES AND 
PASS THROUGH A GIVEN POINT. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. vm. (1867), 
pp. 220—222.] 
Consider the triangles which touch three given lines; the three lines form a 
triangle, and the lines joining the angles of the triangle with the points of contact of 
the opposite sides respectively meet in a point S: conversely given the three lines and 
the point S, then joining this point with the angles of the triangle the joining lines 
meet the opposite sides respectively in three points which are the points of contact 
with the three given lines respectively of a conic; such conic is determinate and unique. 
Suppose now that the conic passes through a given point; the point S is no longer 
arbitrary, but it must lie on a certain curve; and this curve being known, then taking 
upon it any point whatever for the point S, and constructing as before the conic 
which corresponds to such point, the conic in question will pass through the given 
point, and will thus be a conic touching the three given lines and passing through 
the given point. And the series of such conics corresponds of course to the series of 
points on the curve. 
I proceed to find the curve which is the locus of the point S. 
We may take x=0, y = 0, z=0 for the equations of the given lines, and 
x : y : z = 1:1 : 1 for the coordinates of the given point. The equation of a conic 
touching the three given lines is 
a V (x) + b V (y) + c V (z) = 0, 
and the coordinates of the corresponding point A are as — : p : -, that is, taking 
(x, y, z) for the coordinates of the point in question, we have 
a:b:c -fW V(y)' V(*V 
7—2