280 ME. A, CAYLEY’S jVIEMOIE LTOX CAUSTICS. 
IX. 
Consider a ray reflected any number of times at a 
circle ; and let GoG, be the ray incident at G, and GG' 
the last reflected ray, the point at which the reflexion 
takes place or last point of incidence being G. Take 
the centre O of the circle for the origin, and any two lines 
Ow, Oy through the centre and at right angles to each 
other for axes, and let Ox meet the circle in the point A. 
Write 
AO Go — ^05 ^ xGqGj=\Pq 
^ AOG —d ^xGG'^^p 
Z GoG^O=:ip. 
Then the radius of the circle being taken as the centre of the circle, the equation of the 
reflected ray is 
y— sin 6= tan cos ; 
and if there have been n reflexions, then 
— 2(p) = ^o — 2n<p, 
■<P= -4^0 — 212^, 
and therefore the equation of the reflected ray is 
y cos ^sin (- 4 / 0 — 2%<p)4-( — )" sin (^/o — ^o) = 0- 
X. 
If a pencil of parallel rays is reflected any number of times at a circle, then taking 
AO for the direction of the incident rays, we may write ^o=‘P 9 the equation 
of a reflected ray is 
X sin 2n^ -\-y cos 2n(p = ( — )® sin (p. 
Differentiating with respect to the variable parameter, we And 
X COS 2n(p —y sin 2n(p={ — cos (p ; 
and these equations give 
* ‘'>1 
An 1 
(2^^-^- 1 ) cos (2w — I )(p — (2w — I ) cos ( 2 w -f I )(p 1 
(-)” 
y- An ^ 
1^— (2^+1) sin(2w— I)(p+(2w— I) sin (2?^-|-I)(p|, 
which may be taken for the equation of the caustic ; the caustic is therefore an epicy- 
cloid : this is a well-knovm result. 
XI. 
If rays proceeding from a point upon the cii’cumference are reflected any number of 
