434 NOTE ON THE WAVE SURFACE. [281 
is an orthogonal trajectory of the reflected rays. In fact, if 0 be the radiant point, 
and OM' the incident ray, and if from M' we measure off on the reflected ray a 
distance = — OM', that is, on the reflected ray produced backwards, a distance M'P = OM', 
then the whole distance from 0 is OM' — OM' = 0, and the surface which is the locus 
of the point P is consequently an orthogonal trajectory to the reflected rays. But the 
point P may, it is clear, be constructed as follows: viz., on the tangent plane at M' 
let fall the perpendicular OP’, and produce it to a point P such that OP' = P'P. 
And if we produce OM' to M so that OM' = M'M, then it is clear that the locus of 
M is a surface similar and similarly situated with the original surface, but of double 
the magnitude, and that OP is the perpendicular from 0 upon the tangent plane at 
M of the last-mentioned surface. And, by what precedes, the line PM' from P to 
the middle point of OM is a normal of the surface which is the locus of P: the 
corresponding theorem in piano is in fact actually given by Dandelin. 
31s£ Dec., 1858.