On Photometric Paddle-Wheels. 
679 
1910-11.] 
cuts DH at the given value of a m . The necessary vertical and horizontal 
lines appear on the chart ; but the radial lines have not been drawn, as 
greater accuracy can be obtained by stretching a fine black thread in the 
required position. The corresponding elliptic part of the boundary (EB 
in fig. 14) is the curved line from E, which cuts FH at the given value of 
a m — the reason why nothing to the right of FH need be considered will 
appear in the course of the next paragraph. 
§ 6. On what is Necessary to Satisfy both the Foregoing 
Conditions. 
It has just been shown that the beam of light must lie within a bound- 
ary of the shape ABC, fig. 14, and it was shown in § 4 that it must lie 
Fig. 15. 
between two lines through 0 each making an angle with the vertical OA 
equal to tan -1 A m . To comply with both conditions it is evidently necessary 
and sufficient that the beam should be confined within the boundary 
AMON, fig. 15. As before, owing to the symmetry of this boundary, it 
will be sufficient to provide a means of determining AMO, one half of it. 
The parts AM and AO can be found drawn to scale in Chart V., as just 
explained ; and the position of the line OP relative to them, although not 
marked on the chart, can readily be ascertained as follows. DE is of 
unit length, and therefore a line joining D with a point on EF distant 
from E by a length equal to A m would make an angle tan -1 A m with the 
vertical ; i.e. would have the same slope as the required line. If there- 
fore a parallel ruler be laid on the chart with its edge on D and on the 
given value of A m on EF, and then be moved parallel to itself until its 
edge is at the point on DH corresponding to O, the required line 
is found. And this completes the quantitative determination of the 
restricting boundary. 
