158 Proceedings of the Royal Society of Edinburgh. [Sess. 
Equation (1) becomes in the usual way 
where p is independent of r and 0 along any one ray. 
Thus 
leading to the integral 
3T 3T 1 //r 2 
W =P ’ df =± rd\7~ P J ■ ' 
■ ■ (2) 
t-WtVS-") • ■ ■ 
■ (3) 
The equation of the ray is obtained by equating 3T /dp to an arbitrary 
constant, or 
= constant 
0 ) 
In every case the ray is symmetrical with reference to the radius which 
bisects the arc between the source and the point of emergence. This 
radius, which will for the moment be taken as the line of reference, meets 
the ray at its vertex distant from the centre by the stationary or turning 
value of r for each particular ray. Let this stationary radial distance be 
represented hy 0 . Then, integrating from r = z to r — 1, and from 0 = 0 to 
d = a, we find 
(5) 
where 2 a is the arc between the source and the point of emergence. 
Let <p , measured with reference to the radius of symmetry, be the 
angle at which the radius cuts the ray at any given point. Its cotangent, 
sine, and cosine are given by the following expressions — 
cotan </> = 
dr 
rdO 
sin (f> — 
vdQ 
ds 
po 
r 
( 6 ) 
COS cf> = 
dr 
ds 
d-s being the element of arc. 
From the second of these relations by differentiation with regard to r 
there results 
. d<±> pv 
p dv 
r dr ’ 
or, by use of the third and second equations of (6), 
