30 ROYAL SOCIETY OF CANADA 
by the opaque figure, ZL, of the screen: this is the penumbra. Designat- 
ing by J the illumination outside of AM and by q the fraction of the 
diaphragm visible from a point 7 
of the plate, the illumination 7 at 
Das: 
(a gue 

The uncovered portion of the 
diaphragm is found by projecting, 
on the plane of the screen, the 
perspective, abcd, of the diaphragm 
seen from 7. The planes of the 
diaphragm and screen being paral- 
lel, the figure abcd is similar to 
ABCD, and the portion of abed 
which is outside of the opaque fig- 
ure LL, bears to the whole surface, 
abcd, the same ratio as the un- 
covered portion of the diaphragm 
bears to the whole surface, ABCD. 
The visible fraction of the dia- 
phragm is thus ascertained by 
finding the fraction of the diaphragm’s perspective not covered by the 
Opaque part of the screen. 
A line, FG, on the surface of the plate, such that from every one of 
its points the same fraction of the diaphragm is visible, is a line of equal 
illumination. When 7’ is displaced along this line, the uncovered portion 
of the diaphragm may change its shape, but its area remains constant. 
Let P be the centre of the diaphragm and p its perspective on the 
plane of the screen, T being the point of sight. When 7 moves along 
FG, p describes another curve, fg, which is the perspective of G, seen 
from P. The screen and plate being parallel, these curves are similar, and 
any relation existing between the curves described by p holds good for 
the curves described by 7. In process work, the distance from the 
diaphragm to the plate is so large compared to the distance from the 
screen to the plate, that these two sets of curves are practically equal. It 
will be assumed hereafter that they are exactly equal, but this is done 
merely for the sake of simplicity, as the relations which we will find are 
independent of this assumption. According to this, instead of determining 
the lines of equal illumination on the plate, it will be sufficient to calculate 
the curves described on the screen by the perspective of P when the 
visible portion of the diaphragm remains constant. 
We will commence with the simplest case, that of the chess-board 
screen with square diaphragm, and we will suppose both to be so adjusted 


Oe i a ee, See 
Ries 1. 
