]\IE. A. CAYLEY’S MEMOIE UPON CAUSTICS. 
275 
which is equivalent to 
□ QGN'+CiW-^- 1) VQGN®)-n^^' VQ^'=:0, 
and the factors are 
V^GN\/,:^^dQGN'+(^'-1) VQGI?]+ D^GN. VQGN=0 ; 
it is in fact easy to see that these equations represent lines passing through the point G 
and inchned to GN at angles +<p', where (p' is given by the equations 
sin (p=fJtj sin p' 
tan (p= 
VQGN 
□ qgn’ 
and there is no difficulty in distinguishing in any particular case between the refracted 
ray and the false refracted ray. 
In the case of reflexion [/j= — 1, and the equations become 
V^GN. □QGN+DjGN. VQGN=0; 
the equation 
V^GN. dQGN- D^GN. VQGN=0 
is ob\iously that of the incident ray, which is what the false refracted ray becomes in 
the case of reflexion ; and the equation 
V^GN. □QGN+ □<?GN.VQGN=0 
is that of the reflected ray. 
II. 
But instead of investigating the nature of the caustic itself, we may begin by flnding 
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 f/t/“hGQ, 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, 
how'ever, 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 |, ri be the coordinates of the point Q, and 
a, 8 the coordinates of the point G, then |, ri, a, (3 are in efiect functions of a single 
arbitraiy parameter ; and if we write 
GQ"=(^-ccr-h(^-(3r 
2 0 2 
