288 
ME. A. CAYLEY’S MEMOIE UPON" CAUSTICS. 
Avhence 
. fflcoss — ccos« „ csina — asms 
A= — irr-T — > JJ= 
sin (a — e) 
sin (a— e) 
Substituting these values, the equation of the reflected ray becomes 
a sin (2a— g)=M sin (a— £)+c sin {a— 6), 
from which, and its difierential with respect to the arbitrary- parameter a, the equation of 
the caustic or envelope of the reflected rays will be found by eliminating a. 
In this, a being the only quantity treated as variable in the diiferentiation, let 
Therefore 
2a— £=2ip. 
a=^+^(^+s). 
and the equation becomes 
asin2<p=Msin{(p+-|(^— s)} +csin{<p— 1-(6 — s)} 
Make 
P= 
{u + c) cos — e) 
2a 
(m-c) sin !($-£) 
2a 
also 
and the equation becomes 
with the condition 
Hence 
1 1 
x= , y =-. — •> 
cos (p sin (f 
P^+Qy=l, 
'Pz=z'KX~^ 
Q=Xy 
Multiplying by x and y, and adding, we And X=1 ; therefore 
x~^='P'^, y~^=Q^. 
Hence 
pt + Ql=l; 
or restoring the values of P and Q, 
{(m+c) cos [{u—c) sin g)}^=l, 
the equation of the caustic. 
XVHI. 
But the equation of the caustic for rays proceeding from a point and reflected at a 
circle may be obtained by a different method, as follows : — 
Take the centre of the circle for origin ; let c be the radius of the circle, a, b the 
coordinates of the radiant point, a, (3 the coordinates of the point of incidence, x, y the 
coordinates of a point in the reflected ray. Then we have from the equation of the 
