ME. A. CAYLEY’S MEMOIE UPON CAUSTICS. 
281 
times at a circle, then taking the point A for the radiant point, we have ^o=0, 
and the equation of a reflected ray is 
x%m{2n-\-V)<p-\-y cos (2wH-l)ip=( — )”sin <p. 
Difierentiating with respect to the variable parameter, we find 
X cos sin — sin (p ; 
and these equations give 
X=: 
^n<p-\-n cos (2^^+2)(pj, 
which may be taken as the equation of the caustic ; the caustic is therefore in this case 
also an epicycloid : this is a well-known result. 
(-] 
2/i -f- 1 
(?^4-l) cos 27i<p—ncos (2w+2)(p 
XII. 
Consider a pencil of parallel rays refracted at a circle ; take the radius of the circle as 
unity, and let the incident rays be parallel to the axis of x, then if <p, (p' be the angles of 
incidence and refraction, and pb or ^ the index of refraction, so that sin ^'=^sin<p, the 
coordinates of the point of incidence are cos (p, sin <p, and the equation of the refracted 
ray is 
y— sin(p= tsai(<p — ip')(x — cos <p), 
i. e. 
cos {<p—(p'){y— sin <p)= sin {(p—<p'){x— cos <p), 
or 
y cos,{<p—<p')—xmx {<p—<p')z=z sin (p\ 
which may also be written 
{y cos <p— ipsin cos <p^+(?/sin (p-|-;r cos <p — l) sin <^'=0 ; 
or writing k sin <p, \/\ — T^ sin^ instead of sin (p\ cos <p', and putting for shortness 
yco%(p — a^sin <p=Y 
y sin <p-\-x cos ^=X 
k sin <p 
sin^ <p 
lO), 
the equation of the refracted ray becomes 
Y-f(D(X-I)=0. 
And difierentiating with respect to the variable parameter (p, observing that 
dY „ <?X ^ 
df ' d(p 
d^ 
A: cos 9 
cotip 
2 p 
MDCCCLVII. 
