670 
Proceedings of the Royal Society of Edinburgh. 
[Sess. 
i.e. 
sin 
2 
1 . 
J(l + A ' 2 sec 2 a) ; 
( 12 ) 
and as it can easily be shown that in practice e would never exceed about 
4 0, 5, therefore, expanding and keeping only the first power of e, we have 
so that 
and from (11*) 
Let 
then 
U = ^ cot . 
n 2 
X = \/ { (1 - -0314k) 2 “ 1 [ cos a ' 
(13) 
(14) 
(15) 
It appears from the above that the connection between X and p in the 
case of any given paddle-wheel of the types A or B will depend on a, 
the azimuth angle through which the latter is turned. And the position- 
ratios of the rays of the beam are none of them to be greater than X, if the 
beam is to be sensibly homogeneous in intensity. But it is necessary, of 
course, that this homogeneity should exist whatever be the value of the 
angle through which the paddle-wheel is turned. Therefore the position- 
ratios of the rays are none of them to be greater than A m , where \ m is the 
smallest value which X assumes throughout the whole range of azimuth 
angles through which the motor will be swung (say 0 > a ij> a m ). 
\ m is found as follows. In (12) the left-hand member is always a positive 
proper fraction (assuming e 0 < i r), hence its equivalent (1 — e/2, cot ej 2) 
equal to (1 — *0314^/^ . cot ej 2) is always a positive proper fraction, 
hence from (13) A/ is a minimum, i.e. X — \ m , when simultaneously, 
t Q is a min. [ see (11) & (10)] when e 0 is a max. 
is a max.. 
fsee (9)] when a is a max. 
la 
is a max. 
when a = 
Hence from (15) 
A ”* - s / i (1 - - 0 314«,„) 2 1 } cos a ”‘ • 
• ( 16 ) 
