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 denned 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 cöordinates. But, where the cöordinates of the primary curve have to be eliminated, the problem becomes more difficult, and unless this condition be fulfilled the expression cannot be considered a true analytical solution of the locus. In the present paper, differential and algebraic expressions are first found containing the cöordinates of the focal point corresponding to a point in the primary or reflecting curve, and the cöordinates of the latter are then eliminated between the differential and algebraic equations.