On Photometric Paddle-Wheels. 
663 
1910-11.] 
discovering what are the best forms and dimensions for 'photometric 
paddle-wheels generally, and therefore of course it is not required in 
connection with the actual use of such wheels, any more than the elaborate 
calculations relating to the design of a ship’s hull are required in connec- 
tion with the actual navigation of the ship. 
§ 3. The Relation between the Angle of Azimuth of the Axle of 
the Paddle-Wheel and the Corresponding Reduction of the 
Light Intensity. 
We begin by finding an expression for e. It is clear that e may be 
regarded as the angle between two planes passing through the y axis of 
fig. 8 and through the two intersections respectively of the chosen ray of 
Here Oy coincides with the axle of the motor (supposed 
horizontal) ; Ox is horizontal and at right angles to 
Oy ; while Oz (z not shown) is vertical. 
The ray of light AB is parallel to the xy plane, and at a height 
q above it ; and makes an angle a with the yz plane. 
light AB with a cone whose axis is the y axis and whose semi-vertical angle 
is 90° — /3, 2/3 being equal to the angle fOg, fig. 1 . It will be observed that 
we are considering the motor axle to be stationary, while the direction of 
the beam of light is movable, the opposite of what obtains in practice, 
but it is simpler to do this, and the results are of course identical. 
The ray AB is given by its equations, 
z=q (i) 
cos a . x — sin ay =p ..... (5) 
The equation of the cone is 
x 2 — cot 2 /3 . y 2 = - z 2 . . . . . (6) 
The equation of any plane through the y axis is, 0 being the angle it 
makes with the 2 axis, 
x = tan O.z ...... (7) 
If (7) passes through the intersection of (4), (5), and (6), this determines 
