278 Proceedings of the Boyal Society of Edinburgh. [Sess. 
depending on the fixed distance from the plate, and on the original 
(uniform) intensity of the beam, and x measures the distance from the axis 
of those incident rays which after transmission emerge at the angle 0. 
Hence, 
and 
7 xdx 
i — k— — —— . 
sm Odd 
i; 
i sin 6d0 = 
Jk.x = | 2 J i sin Odd j 
(4) 
The analytical form of the function i sin 0 is unknown, only certain 
pairs of corresponding values of i and 0 being given by the experiments. 
Therefore, in order to obtain the values of x it is necessary to plot the 
values of % sin 0 against 0, to draw a fair curve through the points 
obtained, and to integrate graphically from 0 up to each value of 6. But 
equation (3) gives the values of dy/dx corresponding to the values of 6. If, 
therefore, we plot dy/dx against Jk.x, and integrate graphically, we obtain 
Jk.y as a function of Jk.x. This completely determines the shape of the 
cavity ; the only effect of the unknown constant Jk is to leave its actual 
size uncertain. Now it is evident that “ similar ” cavities will give identical 
relative light-distributions ; and hence to find the size of cavity in any 
given case we must know not only the relative distribution, but also the 
ratio of the actual intensity of the incident light to that of the emergent 
light, for some one value of 0. As this was not determined in the foregoing 
experiments, the drawings in fig. 6 show only the shape of the cavities 
equivalent to the plates used in experiments 1 and 8. On comparing 
the relative intensity curves of these two plates, it will be seen that the 
intensity falls off more rapidly with increasing angle in the case of no. 1 
than in the case of no. 8, and hence we should expect that A, the Equivalent 
Cavity of the former, would be flatter near the origin than B, the Equivalent 
Cavity of the latter, but would turn upwards more steeply afterwards, as 
indeed the figure shows to be the case. This point is further brought out 
by a comparison of the curvatures of the two curves at the origin, the radius 
of curvature of A being calculated to be 2335 and of B 15’2, the unit of 
length being a side of one of the large squares of the paper. 
It is of interest to compare with these the particular Equivalent Cavity 
which corresponds to uniform distribution of the light at all angles. Its 
equation is readily obtained as follows. 
