1889 - 90 .] Lord McLaren on Reflexion- Caustics of Curves. 283 
pencil is oblique, or inclined to this axis of symmetry, the rays do 
not intersect at all, except in a principal plane. 
To simplify the investigation the pencil of parallel rays will, in 
the first instance, be supposed to be parallel to the axis of symmetry 
(X) of the reflecting curve, which is also taken as the axis of polar 
coordinates. The solutions are then extended to the case of oblique 
pencils. In the notation of this paper, 
r, 0 are the coordinates of the primary, or reflecting curve ; 
p, to are the coordinates of its pedal ; 
v, \ Jr are the coordinates of its negative pedal ; 
B, 0 are the coordinates of the caustic. 
x, y ; X, Y are rectangular coordinates of the primary curve and 
its caustic. 
p is the radius of curvature of the primary curve, and is the only 
radius of curvature necessary to be considered. 
t is the angle of incidence or reflexion. 
v is the angle between any radius-vector and the perpendicular 
from the pole on the tangent. 
In the fundamental relation (1) we have a variable origin (the 
incident point of the particular ray), and a reference line, p (the 
radius of curvature), whose direction varies from point to point of 
the reflecting curve. The transposition to a definite origin and a 
fixed reference line (X) may be effected by finding an expression 
for the difference of position of a point (S) of the reflecting curve 
and a corresponding point (C) of the caustic, in coordinates, x, y , 
of the reflecting curve, and X, Y of the caustic. 
Observing that the inclination of the reflected ray to the axis of 
X is 2i, we have, evidently, from (1) 
If p be the perpendicular drawn from the origin to the tangent 
then p is parallel to the normal at S ; and, as the incident ray is 
assumed parallel to the axis of reference, we have, 
at any incident point, S 1? and o> be the inclination of p to the axis, 
O) = i 
(3a) 
