On Photometric Paddle-Wheels. 
677 
1910-11.] 
within the boundary ABC, fig. 14 (only the ujpper halves of the projections 
of the discs are shown, because of course the beam is always entirely above 
the axle). 
The beams of light employed in photometry have as a rule either 
circular or rectangular sections. Hence, what concerns one in practice is the 
question of what is the largest size of circle, or of rectangle (with a given 
ratio of sides), that can be included within the fig. ABC. To facilitate the 
finding out of this, Chart V. has been constructed, in which for a number of 
different values of the determining variables c and a the restricting 
boundary ABC is drawn to scale. Or, rather, the essential parts of ABC 
are drawn to scale, for evidently it will be sufficient to give half of it — 
E 
namely, one of the elliptic sides AB, and the centre line AO drawn in the 
proper position relative to AB. 
The restriction on the size of the beam will be most severe when 
a is a maximum ; hence, it will be enough to consider the case of a = a m , 
where a m is the largest azimuth angle through which the paddle-wheel 
will be turned. 
On considering the way in which fig. 14 is obtained, it will be clear that 
the semi-major-axis DE must always be unity, because the radius of the 
paddle-wheel is taken as unity * ; and that the distance DO of the centre 
line AO, from DE, must be equal to c sin a m . In Chart V. (where DE 
corresponds to the DE of fig. 14) this distance can be found as follows. 
The values of a m marked off along the line DH are placed at distances from 
D equal to sin a m ; and along ED are marked off values of c equal to their 
distances from E. It follows that the required centre line, at a distance 
c sin a m from DE, must pass through the intersection of the horizontal line 
which cuts ED at the given value of c, with a radial line from E which 
* See the final sentence of the “Definitions” of § 1. 
