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Proceedings of the Koyal Society of Edinburgh. [Sess. 
able to obtain some experimental information, but meanwhile I have con- 
tented myself with calculating what may be termed the “ equivalent 
cavity ” of the pits for two of the pieces of glass. That is to say, I have 
calculated, and plotted in fig. 6, the particular shape of pits which would 
give the same distribution of scattered light as do these two plates, 
assuming the pit to be a surface of revolution with its axis of rotation 
normal to the surface of the glass. The calculation is made as follows 
In fig. 7 is shown a section of the glass plate, the x axis coinciding 
with the smooth surface and PQR being the equivalent cavity. If a beam 
of rays parallel to the y axis fall on the cavity, the path of the ray, which 
emerges from the plate at the angle 0, can be ascertained thus. Let the 
ray be SQTW. Draw any arbitrary fixed line AB above the cavity, 
parallel to the x axis and at a distance a from it. Then if Q be (x, y) and 
T (x' } 0), and the refractive index of the glass be ju , ; and if the ray produced 
backwards from T meet at N a line through the origin which makes an 
angle 0 with the x axis, the optical length X of the path from S to the 
virtual point N is given by the equation, 
A = a-y + n{y 2 + (x - x) 2 y - x sin 0. 
But the positions of the variable points Q and T must be such that X is 
a minimum. 
Hence 
and 
= sin 6 . . . (2) 
. , yf- - {x - x) 
dX _ o . i e dy _ dx 
dx ; dx ^ J {y 2 + (x' - x) 2 } 
dX _ # . x — x 
dx ; ^ J{y 2 + (x - x) 2 } 
dy sin 0 
dx J(fj. 2 - sin 2 9) - 1 
(1) and (2) lead to 
( 3 ) 
