310 
ME. A. CAYLEY’S MEMOIE UPOX CALSTICS. 
XXXIX. 
Again, to compare the general equation mth that previously obtained for parallel rays 
refracted at a circle, we must write c=l, «=oo , Z=^ (for the equation of the 
refracted ray was taken to be 'Kj]c-\-Yy ■\-k—0 ) ; we have then 
and after the substitution «=co . The equation becomes in the first instance 
-«‘(l+i=+W+2FfflX)=0 ; 
and then putting a=co , or, what is the same thing, attending only to the terms which 
involve and throwing out the constant factor we obtain 
(X^+Y^XX^-1-A:7-4^^Y^=0, 
or 
XXX^-l-^7+YXX+l+^}(X-l--^)(X+l-/?:XX-l-^’)=0. 
which agrees with the former result 
XL. 
It was remarked that the ordinary construction for the secondary caustic could not 
be applied to the case of parallel rays (the entire curve would in fact pass ofi" to an 
infinite distance), and that the simplest course was to measure the distance GQ from a 
line through the centre of the refracting circle perpendicular to the direction of the 
rays. To find the equation of the resulting curve, take the centre of the circle as the 
origin and the direction of the incident rays for the axis of x ; let the radius of the circle 
be taken equal to unity, and let denote, as before, the index of refraction. Then if a, /3 
are the coordinates of the point of incidence of a ray, we have a^+/3^=l, and consider- 
ing a, j3 as variable parameters connected by this equation, the required curve is the 
envelope of the circle. 
Write now a=cos^l, (3=sin^, then multiplying the equation by —2, and UTiting 
1+ cos 2^ instead of 2 cos^ the equation becomes 
Ifi- cos2^—2(jt/^(x^-\-y^—2xcos S—2y sin ^+1)=0, 
which is of the form 
A cos 2^+B sin 2^-|-C cos sin ^+E=0, 
