292 
ME. A. CAYLEY’S iLEMOIE LTOX CAUSTICS. 
The tangents parallel and perpendicular to the axis of x are most readily obtained 
from the equation of the reflected ray, viz. 
. _ A , -I \ cos 23 — cos 9 , „ 
(-2«cos^+l)^+ y+a=0; 
the coefficient of x (if the equation is first multiplied by sin d) vanishes if sin &= 0, -which 
gives the axis of x, or if cos which gives y= + 
to the axis of x. 
The coefficient of y vanishes if a cos 2^ — cos ^=0 ; this gives 
for the tangents parallel 
cos 
5 sin^=^(4(Z^— l+\/8a^+l): 
4(2 
and the tangents perpendicular to the axis of x are given by 
— 2a 
x=-^ 
1+ \/8aHl 
these tangents are in fact double tangents of the caustic. In order that the point of 
contact may be real, it is necessary that sin 6, cos 6 should be real ; this -will be the case 
for both values of the ambiguous sign if a>OT =1, but only for the upper value if a<l. 
It has just been shown that for the tangents parallel to the axis of x^ we have 
y=± 
2a 
the values of y being real for a>\: it may be noticed that the value i is 
greater, equal, or less than, or to y=2cv‘\/\—a^, according as a>= or this 
depends on the identity (4a^ — 1) — I6a®(I — a^)=(2a^—iy(2a^-\-l). 
To find the points of intersection -with the reflecting circle, x^-\-y ^ — 1 = 0, we have 
(3a^-l-2axy-27a%l-af)(l-aJ=0; 
or reducing 
8<zV+(— 27«^-|-18a^— I5)«V+(54(2^— 36a^+6)a.r+(— 27a^+I8a^+I)=0, 
i. e. 
{ax—iy{%ax—27a*-\-l^a?-\-l)=^. 
The factor {ax—\y equated to zero shows that the caustic touches the chcle in the 
points y='ii\/ 1”^? points in which the circle is met by the polar of 
the radiant point, and which are real or imaginary according as a> or <1. The other 
factor gives 
x= 
27ffl^ — 18a^-l 
8a 
Putting this value equal to +1, the resulting equation is (a+I)(27«‘+9«-l-I)=0, and 
it follows that x will be in absolute magnitude greater or less than 1, i. e. the points in 
question will be imaginary or real, according as a>I or a<I. 
