1889-90.] Lord McLaren on Reflexion-Caustics of Curves . 281 
On the Reflexion-Caustics of Symmetrical Curves. 
By the Hon. Lord M‘Laren. (With Two Plates.) 
(head April 7, 1890.) 
The caustics here considered are those resulting from the reflexion 
of systems of parallel rays. The subject is purely geometrical. 
The caustic is considered as a derived curve, and is defined as the 
envelope of a system of lines drawn from the primary curve, whose 
inclination to a given line is double the inclination of the normal 
at the incident point. 
It is comparatively easy to find an expression for a caustic in 
mixed coordinates. But, where the coordinates of the primary 
curve have to be eliminated, the problem becomes more difficult, 
and unless this condition be fulfilled the expression cannot be con- 
sidered a true analytical solution of the locus. In the present 
paper, differential and algebraic expressions are first found contain- 
ing the coordinates of the focal point corresponding to a point in 
the primary or reflecting curve, and the coordinates of the latter 
are then eliminated between the differential and algebraic equations. 
So far as I am aware, very little has been written on the caustics 
of reflexion for parallel rays. 
Professor Cayley’s investigations relate to the caustics of con- 
verging and diverging pencils, and include refraction-caustics. In 
the recent paper of Mr Mannheim, published in the Transactions of 
the Academia dei Lincei , the author develops a method of deter- 
mining a series of points which are the foci of the reflected rays for 
a given curve or surface; but his method does not furnish the 
equation of the continuous caustic or locus of focal points which I 
conceive to be geometrically the only admissible solution. I may 
add that a method analogous to that here employed was used by 
Mr Childe, in his Treatise on Reflected Ray-Surfaces and their Rela- 
tion to Plane Reflected Caustics , to determine the caustic by reflexion 
for a radiant point which is coincident with the pole or origin of 
polar coordinates of the reflecting curve. Prom the examples given 
by Mr Childe of the application of his method, it is evidently 
applicable generally to two-term polar equations ; though the author 
does not directly announce this limitation, nor does he treat at all 
