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XIII. A Memoir upon Caustics. By Arthur Cayley, Esq. 
Eeceived May 1, — Eead May 8, 1856. 
The following memoir contains little or nothing that can be considered new in principle ; 
the object of it is to collect together the principal results relating to caustics in piano., the 
reflecting or refi:acting curve being a right line or a circle, and to discuss with more care 
than appears to have been hitherto bestowed upon the subject, some of the more remark- 
able cases. The memoir contains in particular researches relating to the caustic by refrac- 
tion of a circle for parallel rays, the caustic by reflexion of a circle for rays proceeding 
from a point, and the caustic by refraction of a ch’cle for rays proceeding from a point ; the 
result in the last case is not worked out, but it is shoAvn how the equation in rectangular 
coordinates is to be obtained by equating to zero the discriminant of a rational and 
integral function of the sixth degree. The memoir treats also of the secondary caustic 
or orthogonal trajectory of the reflected or refracted rays in the general case of a reflecting 
or refracting circle and rays proceeding from a point ; the cuix^e in question, or rather a 
secondary caustic, is, as is well known, the Oval of Descartes or ‘ Cartesian’ : the equation 
is discussed by a method which gives rise to some forms of the curve Avhich appear to have 
escaped the notice of geometers. By considering the caustic as the evolute of the secondary 
caustic, it is shoAvn that the caustic, in the general case of a reflecting or refracting circle 
and rays proceeding from a point, is a curve of the sixth class only. The concluding part of 
the memoir treats of the curve which, when the incident rays are parallel, must be taken 
for the secondary caustic in the place of the Cartesian, which, for the particular case in 
question, passes off to inflnity. In the course of the memoir, I reproduce a theorem 
first given, I believe, by me in the Philosophical Magazine, viz. that there are six 
difierent systems of a radiant point and refracting circle which give rise to identically the 
same caustic. The memoir is dhided into sections, each of which is to a considerable 
extent intelhgible by itself, and the subject of each section is for the most part explained 
by the introductoiy paragraph or paragraphs. 
I. 
Consider a ray of light reflected or refracted at a curve, and suppose that |, 7\ are the 
coordinates of a point Q on the incident ray, a, /3 the coordinates of the point G of 
incidence upon the reflecting or refracting curve, «, h the coordinates of a point N upon 
the normal at the point of incidence, y the coordinates of a point q on the reflected or 
refracted ray. 
2 0 
MDCCCLVII. 
