336 
145 
145. 
A MEMOIR UPON CAUSTICS. 
[From the Philosophical Transactions of the Royal Society of London, vol. cxlvii. for the 
year 1857, pp. 273—312. Received May 1,—Read 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 refracting 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 remarkable cases. The memoir contains in particular researches 
relating to the caustic by refraction 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 circle for rays proceeding from a point; the result in the last case is not 
worked out, but it is shown 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 curve 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 which 
appear to have escaped the notice of geometers. By considering the caustic as the 
evolute of the secondary caustic, it is shown 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 infinity. 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 different systems of a radiant point