ME. A. CAYLEY’S MEMOIE UPON CAUSTICS. 
287 
and representing this by 
we have 
equations which give 
and consequently 
’ sin'-* 
X=^cos(p— v^l — F 
Y=ksin(p-\- cot sin^ (p, 
X cos <p-\-Y sin (p=k 
X2_|_Y* 
’ sin^ <p 
and we have 
which gives 
^ \/X^ + Y^-l 
cos <P= .,, ' ■■■. , 
^ V/X2+Y2 
xy X^ + Y^- 1 + Y --^;V'X=' + = 0, 
(X^+Y^)(X^-l-A;^)=-2/&Y-v/X^+Y^; 
or dividing out by the factor the equation becomes 
yx^+Y^(X^ -l~k^)=- 2kY, 
from which 
(X^+Y^)(X^-1-F)^-4FY^=0 ; 
or reducing and arranging, we obtain 
X^(X^-l-y^;^)^+Y^(X+l+/i;XX+l-/i:XX-l+/rXX-l-A:)=0 
for the tangential equation of the caustic by refraction of a circle for parallel rays. The 
caustic is therefore of the class 6. 
XVII. 
Suppose next that rays proceeding from a point are reflected at a circle. 
A very elegant solution of the problem is given by Lageange in the Mem. de Turin ; 
the investigation, as given by Mr. P. Smith in a note in the Cambridge and Dublin 
Mathematical Journal, t. ii. p. 237, is as follows: — 
Let B be the radiant point, BBP an incident ray, and PS a reflected ray ; CA a flxed 
radius; ACP=a, ACB=£, reciprocal of CB=c, reciprocal of CP=«. The equations of 
the incident and reflected ray, where may be written 
w=:A sin ^-fB cos 6 incident ray, 
«i=A sin (2a— ^)+B cos (2a— reflected, 
the conditions for determining A and B being 
a=A sin afl-B cos a 
c =A sin £ +B cos £, 
