282 Proceedings of Royal Society of Edinburgh. [sess. 
of the caustics by reflexion of pencils of parallel rays. It appears to 
me that the case investigated by Mr Childe, and the case of parallel 
rays here treated, embrace all that can be found by the method of 
auxiliary curves, and my paper may therefore be regarded as comple- 
mentary to this part of Mr Childe’s work, although the auxiliary 
curves used and the proofs are quite different for the two cases. 
The general solution here given (which is essentially geometrical) 
was suggested by my solution of a particular case, that of the 
Reflexion-Caustic of the Parabola, which appeared in the Monthly 
Notices of the Royal Astronomical Society for 1887. The solution 
is general for the parallel-ray caustic of any reflecting curve which 
can be expressed as a two-term polar equation. The equivalents of 
the polar equations in Cartesian coordinates are then determined for 
the curve of any degree n ; and it is shown that these include funda- 
mental forms or types of every class of symmetrical algebraic curves, 
whether these be finite or infinite in extent, central or parabolic. 
The solution may be held to apply generally to algebraic curves of 
perfect symmetry , if we consider those curves only to have perfect 
symmetry whose axes of symmetry are separated by equal angular 
intervals ; in other words, curves whose branches are all equal and 
are symmetrically disposed about a centre. 
1. Differential Equation of a Reflected Ray. 
I begin with the fundamental expressions for the length (L) of a 
reflected ray, from the incident to the focal point. These are 
L = J/Oi c °st . . . (1); L = Jp 2 secfc . . . (2), 
where p 15 p 2 are respectively the greatest and least radii of curvature 
at the incident point, and t is the angle of incidence or reflexion. 
For surfaces of revolution, to which these investigations are 
usually confined, the relation (1) applies to the focus of consecutive 
reflected rays in a principal plane ; and the relation (2) applies to 
the focus of consecutive rays reflected from a cyclic section of the 
reflecting surface (see “ Theory of Systems of Rays,” Trans. Royal 
Ir. Ac., vol. xv. p. 97). I shall deal only with the form (1) applicable 
to reflexion in a principal plane, it being evident that the cone of 
rays represented by (2) has its vertex or focus in the axis of the 
surface of revolution if the incident pencil is direct, and that if the 
