372 
A MEMOIR UPON CAUSTICS. 
[145 
the equation becomes 
\x 2 + y 2 + (ab + ac+ ad + bc + bd + cd)} 2 
— (abc + abd + acd + bed) x 
— | (ash 2 + a 2 c 2 + a 2 d 2 + b 2 c 2 + b 2 d 2 + c 2 d 2 — (kibed) = 0, 
which is of the form 
(ix 2 + y 2 — a) 2 + 16.4 (x — m) = 0, 
and, as already remarked, signifies that the fourth power of the tangential distance of 
a point in the curve from a given circle, is proportional to the distance of the same 
point from a given line. The circle in question (which may be called the dirigent 
circle) has for its equation 
x 2 + y 2 + \ (ab + ac + ad + be + bd + cd) = 0 ; 
the line in question, which may be called the directrix, has for its equation 
a 2 b 2 + a 2 c 2 + a 2 d 2 + b 2 c 2 + b 2 d 2 + e 2 d 2 — Gabed _ 
rjQ I — (J • 
4 (abc + abd + acd + bed) 
the multiplier of the distance from the directrix is 
abc + abd + acd + bed. 
It may be remarked that a, b, c, d being real, the dirigent circle is real; the equation 
may, in fact, be written 
x 2 + y 2 = i [(a + b) 2 + (a + cf + (a + d) 2 -f (b 4- c) 2 4 (b + d) 2 + (c + cZ) 2 ]. 
XXXIV. 
Considering the equation of the Cartesian under the form 
(x 2 + y 2 — a) 2 + 16A (x — to) = 0, 
the centre of the dirigent circle x 2 + y 2 — a = 0 must be considered as a real point, 
but a may be positive or negative, i.e. the radius may be either a real or a pure 
imaginary distance: the coefficients A, m must be real, the directrix is therefore a real 
line. The equation shows that for all points of the curve x — m is always negative 
or always positive, according as A is positive or negative, i.e. that the curve lies 
wholly on one side of the directrix, viz. on the same side with the centre of the 
dirigent circle if A is positive, but on the contrary side if A is negative. In the 
former case the curve may be said to be an ‘ inside ’ curve, in the latter an ‘ outside ’ 
curve. If m = 0, or the directrix passes through the centre of the dirigent circle, 
then the distinction between an inside curve and an outside curve no longer exists. 
It is clear that the curve touches the directrix in the points of intersection of this 
line and the dirigent circle, and that the points in question are the only points of 
intersection of the curve with the directrix or the dirigent circle; hence if the 
directrix and dirigent circle do not intersect, the curve does not meet either the 
directrix or the dirigent circle.