145] 
A MEMOIR UPON CAUSTICS. 
339 
II. 
But instead of investigating the nature of the caustic itself, we may begin by 
finding the secondary caustic or orthogonal trajectory of the refracted rays, i.e. a curve 
having the caustic for its evolute; suppose that the incident rays are all of them 
normal to a certain curve, and let Q be a point upon this curve, and considering 
the ray through the point Q, let G be the point of incidence upon the refracting 
curve; then if the point G be made the centre of a circle the radius of which is 
fxr 1 . GQ, the envelope of the circles will be the secondary caustic. It should be 
noticed that, if the incident rays proceed from a point, the most simple course is to 
take such point for the point Q. The remark, however, does not apply to the case 
where the incident rays are parallel; the point Q must here be considered as the 
point in which the incident ray is intersected by some line at right angles to the 
rays, and there is not in general any one line which can be selected in preference 
to another. But if the refracting curve be a circle, then the line perpendicular to 
the incident rays may be taken to be a diameter of the circle. To translate the 
construction into analysis, let £, 77 be the coordinates of the point Q, and a, /3 the 
coordinates of the point G, then f, 77, a, /3 are in effect functions of a single 
arbitrary parameter; and if we write 
GQ 2 = (Z-«y + (v~/3) 2 , 
Gq 2 = 0 - a) 2 + (y - (3f, 
then the equation 
y?Gq 2 - GQ 2 = 0, 
where x, y are to be considered as current coordinates, and which involves of course 
the arbitrary parameter, is the equation of the circle, and the envelope is obtained 
in the usual manner. This is the well-known theory of Gergonne and Quetelet. 
III. 
There is however a simpler construction of the secondary caustic in the case of 
the reflexion of rays proceeding from a point. Suppose, as before, that Q is the 
radiant point, and let G be the point of incidence. On the tangent at G to the 
reflecting curve, let fall a perpendicular from Q, and produce it to an equal distance 
on the other side of the tangent; then if q be the extremity of the line so produced, 
it is clear that q is a point on the reflected ray Gq, and it is easy to see that 
the locus of q is the secondary caustic. Produce now QG to a point Q' such that 
GQ' = QG, it is clear that the locus of Q' will be a curve similar to and similarly 
situated with and twice the magnitude of the reflecting curve, and that the two 
curves have the point Q for a centre of similitude. And the tangent at Q' passes 
through the point q, i.e. q is the foot of the perpendicular let fall from Q upon 
the tangent at Q'; we have therefore the theorem due to Dandelin, viz. 
43—2