435 
of Edinburgh, Session 1870 - 71 . 
The problem was: to construct a reflecting surface from which 
rays, emitted from a point, shall after reflection diverge uniformly, 
but horizontally . Using the ordinary property of a reflecting sur¬ 
face, we easily obtain the first written equation. By Hamilton’s 
grand “Theory of Systems of Rays,” we at once write down the 
second. 
The connection between them is easily shown thus. Let and 
r be any two vectors whose tensors are equal, then 
a + = 1 + 2gr ~i + ( gT -i)» 
= 2*t-'(1 + S*rr-'), 
whence, to a scalar factor pres, we have 
Hence, putting = U(/3 + aYap ) and r = Up, we have from the 
first equation above 
S. dp [Up + U (J3 -f aYap) ] = 0 . 
But 
d (J3 + aYap) = aYadp = — dp — aSa dp , 
and 
S . a{(3 + aYap) = 0 , 
so that we have finally 
S . dpUp — S . d(/3 + aVap)U(/3 -f- aYap) = 0 , 
which is the differential of the second equation above. A curious 
particular case is a parabolic cylinder, as may be easily seen 
geometrically. The general surface has a parabolic section in the 
plane of a, (3 ; and a hyperbolic section in the plane of /3, a(3. 
It is easy to see that this is but a single case of a large class of 
integrable scalar functions, whose general type is 
the equation of the reflecting surface; while 
S(cn — p)da~ = 0 
is the equation of the surface of the reflected wave: the integral 
