280 
Proceedings of tlie Royal Society of Edinburgh. [Sess. 
(a) 
(b) 
.(«) 
These lead to 
i = r 2 cos 0 
dy x 
dx J(r 2 - a? 2 ) 
cZy sin 0 
dx 
7(2*25 -sin 2 0)-l 
xdx 
sin OdO 
3-25 7(2-25 - sin 2 #) - 4*5 + sin 2 # 
7(2-25 - sin 2 0){3-25 - 2 7(2’25 - sin 2 #)} 5 
or, if we put sin <p — § sin we have the equation 
J | /constant') cos fld ’ 5 ~ CQS M 008 </> - ’667) 
( ' 6 cos c/>(l"083 - cos </>) 2 ’ 
which is adapted for logarithmic computation. 
This leads to 
Angle — d 
0 
2 5 
5 
10 
15 
20 
25 
30 
Relative intensity in that direction 
100-00 
99-12 
95-71 
84-27 
69-18 
53-80 
39-28 
29-21 
As before, the relative total emission is obtained by multiplying the 
above function by sin 0 ; this leads to 
Angle = 6 
0 
2-5 
5 
10 
15 20 
25 
30 
Relative total emission . 
100-00 
23'34 
45-02 
78-98 
96-64 99-30 
89-60 
78-81 
and it can be shown by a method of successive approximation that the 
maximum total emission occurs at the angle 0 = 18°*4. The dissimilarity of 
these results — given by a hemispherical cavity — from all the experimental 
ones can be seen at a glance by comparing its graphs, nos. 15 in fig. 6,* 
with the earlier ones. It is at once apparent that the actual cavities must 
depart from the shape of true spheres by having flatter bottoms and steeper 
sides. And, indeed, on consideration this is what we should expect, because, 
were we to have a plate the surface of which was covered with small 
hemispherical cavities packed together as closely as possible, there would 
still be a large area of the plate in its original flat condition ; and assuming, 
as seems reasonable, that this is not likely to be the case in practice, if we 
* Curve no. 14 in fig. 6 shows the light intensity distribution that would be given by an 
infinitely long semicylindrical cavity, the. axis of the circular cylinder being in the surface 
of the plate. Evidently this case can be calculated from the same equations (a) (6) (c) as 
before, if (c) be changed to i = dx/dd. The results of the calculation are given in Table II. 
no. 14. 
