338 
A MEMOIR UPON CAUSTICS. 
[145 
we obtain 
fG* VQGN* - p* QG\ qGN 2 = 0, 
an equation which is rational of the second order in x, y, the coordinates of a point 
q on the refracted ray; this equation must therefore contain, as a factor, the equation 
of the refracted ray; the other factor gives the equation of a line equally inclined 
to, but on the opposite side of the normal; this line (which of course has no physical 
existence) may be termed the false refracted ray. The caustic is geometrically the 
envelope of the pair of rays, and for finding the equation of the caustic it is 
obviously convenient to take the equation of the two rays conjointly in the form 
under which such equation has just been found, without attempting to break the 
equation up into its linear factors. 
It is however interesting to see how the resolution of the equation may be 
effected; for this purpose multiply the equation by NG 2 , then reducing by means of 
a previous formula, the equation becomes 
('VqGN 2 + nfGl?)VQGN* - p%VQGN* + □ ~QGN*)VqGN* = 0, 
which is equivalent to 
VfGN* (p* □QGN* + (y? -1 )VQM 2 ) - □qGN*VQGN* = 0, 
and the factors are 
VqGN\/p* OQGN 2 + — 1)V QGN* + OqGN. \7QGN = 0; 
it is in fact easy to see that these equations represent lines passing through the 
point G and inclined to GN at angles ± 0', where 0' is given by the equations 
sin <j) = p sin </>', 
tan 0 = 
VQGN 
OQGN’ 
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 p = — 1, and the equations become 
VqGN. OQGN + OqGN. WQGN= 0; 
the equation 
VqGN. OQGN-OqGN. VQGN = 0 
is obviously that of the incident ray, which is what the false refracted ray becomes 
in the case of reflexion; and the equation 
V qGN . □ QGN + OqGN . V QGN = 0 
is that of the reflected ray.