288 
MR. R. S. HEATH ON THE DYNAMICS OF 
r 
The sides of this triangle of reference may be called the principal axes of the conic. 
The condition that an equation of the second degree should represent two straight 
lines, is that the discriminant should vanish. If we consider the equation 
s=% 2 +r+2 2 )> 
and make the discriminant vanish, we get three pairs of lines, which are the repre¬ 
sentatives of the asymptotes in ordinary geometry. If we take the form of the 
equation referred to the principal axes, we see that a pair of these lines passes through 
each angular point of the triangle. The asymptotes, however, no longer touch the 
conic, but are the six lines joining the four points of intersection of the conic with 
the absolute. 
11. If the equation to a conic be 
fr+r, ~r =0 
the tangential equation becomes 
ct~l~ +6 2 m 2 + c 2 n~ =0. 
The tangential equation of the absolute is 
n~ = 0 . 
The foci of a conic may be defined to be the six points of intersection of the 
common tangents to the conic and the Absolute. Confocal conics are those which 
have the same common tangents with the Absolute. Hence the tangential equation 
of a system of confocal conics is 
aH 3 + h-ru~ -fi cVr -f- X(l' 2 -{- m~ -j-« 2 ) = 0 
and therefore, in point coordinates, the equation to a system of confocal conics is 
0 0 O 
—-— -J-——— — o. 
ci~ b~ T- A. c" A. 
Thus confocal conics have the same principal axes. Also it may be shown in the 
usual way that confocal conics cut at right angles, and that two confocals can be drawn 
through any point in the plane. 
12. Ihe equation to a circle whose centre is (/, m, n), and the cosine of whose 
radius is r, is 
lx-\-my-\-xz—r ; 
or, making it homogeneous, 
(lx+my-\-nzy=)°(x 2 +y~ + z~). 
