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 w and 
r be any two vectors whose tensors are equal, then 
L±^ y = i + 2„ t -‘ + c^T-'y 
= 2=t-*( 1 + Ss,T-‘), 
whence, to a scalar factor pres , we have 
( 0 ‘- 
T -h ' 
Hence, putting w = U (j8 + aV ap) and r = Up, we have from the 
first equation above 
But 
and 
S.dp[Up + U(/? + aVap)] = 0. 
d (/3 aV ap) = aVa dp = — dp — aSa dp , 
S . a(/3 + a V ap) = 0 , 
so that we have finally 
S . dpUp — S . d(/3 + aVap)U(/3 + 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 /?, 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 
s.dpfffy P =o, 
the equation of the reflecting surface ; while 
S(a^ — p)dcr = 0 
is the equation of the surface of the reflected wave : the integral 
