On Photometric Paddle-Wheels. 
669 
1910 - 11 .] 
Consider a beam of light transmitted by a paddle-wheel of type A 
or B. As we pass outwards from the middle of the beam towards either 
side of it, the position-ratio of the rays successively encountered goes on 
numerically increasing, and at the same time (8) shows that simultaneously 
the value of e will go on decreasing, and (11) and (10) show that the value of 
t will go on increasing. Therefore, after a beam of light has passed through 
the wheel, the greatest difference in the values of the transmission coefficients 
of the various rays will be found, between the coefficients of the medial 
rays, for which X is zero, and the coefficients of the outermost rays for 
which X has its greatest value ; and consequently, if the difference between 
these coefficients be sufficiently small, the arrangement will be satisfactory. 
If then p per cent, be the largest variation that can be permitted in the 
value of the transmission coefficient, the condition for equality of illumina- 
tion of the rays of the beam is simply that the cross section of the latter 
should be so restricted that no ray has a position-ratio greater than X", 
where X' is that value of X which makes the value of t the transmission 
coefficient of the corresponding rays differ by p per cent, from t 0 , the 
transmission coefficient of the “ medial rays.” 
X' is found as follows : — 
From (L0) 
where E is in radians ; therefore, from (11) 
^ _ j. _ W (E + e o) . 
Also 
therefore 
hP _ n { 
100 2t r 
P’o- e )=7r > 
Ji7T 
£ = - 0628 ^ 
(11*) 
n 
We have also 
and 
. e 
sin — = sin 
c tan a 
2 
+ A' 2 sec 2 a) ’ 
J( 1 + X' 2 sec 2 a) ; 
therefore 
