ME. A. CAYLEY’S MEMOIE UPON CAUSTICS. 
305 
which is of the form 
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 diligent circle) 
has for its equation 
oc^’\-y^+^{ab-\-ac-\-ad-\-hc-\-hd-\-cd)=^. 
The hne in question, which may be called the directrix, has for its equation 
+ c^d'^ — Qabcd „ 
4i{abc+abd+acd-{-bcd) ’ 
the multiplier of the distance from the directrix is 
abc + abd + acd + bed. 
It may be remarked that «, b, c, d being real, the dirigent circle is real ; the equation 
may, in fact, be written 
XXXIV. 
Considering the equation of the Cartesian under the form 
{x^-{-y'^—ccf-\-lQA{x—m) = Q, 
the centre of the dirigent circle — a=0 must be considered as a real point, but a 
may be positive or negative, ^. 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 emwe x—m is always negative or always positive, 
according as A is positive or negative, ^. 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 contrar}' 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. 
XXXV. 
To discuss the equation 
«)^+ 16 A(a;— m)= 0 , 
I write first ?/=0, which gives 
— 2a.2^+16A^4-a'^— 16A?71.=0 
2 s 
MDCCCLVII. 
