]\rR. A. CAYLEY’S MEMOIE UPON CAUSTICS. 
277 
incident rays are the tangents of the caustic which is the last of the series of images ; 
the problem is not in effect different from that of finding the caustic for refraction in the 
case where the incident rays are the tangents to the caustic after the first refraction, but 
the line at which the second refraction takes place is arbitrarily situate with respect to the 
caustic. Thus e. g. suppose the incident rays proceed from a point, the caustic after the 
first refraction is, it will be shown in the sequel, the evolute of a conic ; for the complete 
theory of the combined reflexions and refractions of the pencil by a plate or prism, it 
is only necessary to find the caustic by refraction, where the incident rays are the 
normals of a conic, and the refracting line is arbitrarily situate with respect to the conic. 
V. 
Suppose that rays proceeding from a point 
Q are refracted at a line ; and take the 
refracting line for the axis of y, the axis of 
X passing through the radiant point Q, and 
take the distance QA for unity. Suppose that 
y 
G 
A 
^ Q 
A 
X 
the index of refraction g. is put equal to A 
K 
Then if be the angle of incidence and (p' the 
angle of refraction, we have sin (p'=k ship, and the equation y—x tan®'= tan p of the 
refracted ray becomes, putting for p' its value, 
k sin p 
y— 
--X — tan p=0. 
sin^ 
Differentiating with respect to the va,riable parameter and combining the two equations, 
we obtain, after a simple reduction. 
COS^ip 
sin^ p 
cos® p 
where k' = yr — k'^, hence eliminating 
{kxf-(iyyf=l, 
which is the equation of the caustic. When the refraction takes place into a denser 
medium k is less than 1, and k'^ is positive, the caustic is therefore the evolute of a 
hyperbola (see fig. 1) ; but when the refraction takes place in a rarer medium k is 
greater than 1, and k'^ is negative, the caustic is therefore the evolute of an ellipse (see 
fig. 2). These results appear to have been first obtained by Geegonne. The conic 
(hj'perbola or ellipse) is the secondary caustic, and as such may be obtained as follows. 
VI. 
The equation of the variable circle is 
x^-h{y— tan pf—k^sec^p^O; 
