32 ROYAL SOCIETY OF CANADA 
The curve of equal illumination is therefore a circle, having its centre 
in the middle of the transparent square of the screen. The relation holds 
good so Jong as the portions cut off from : 
the diaphragm are right angle triangles ; 
that is, so long as P remains within the 
square, ABCD, Fig. 2. When outside, as 
in Fig. 4, the visible portion of the dia- 
phragm becomes 
a + 2xy. 
and the illumination, 
7 Ole -- 2xy 
(ea ANS 
20° 
ot (0) il 
Dee —;). 
The curve of equal illumination is an hyperbola, with the sides of the 
hence : 

transparent square as asymptotes. 
This last formula does not change when 
P comes into the opaque square, as in Fig. 5; 
the curve of equal illumination is still an 
hyperbola, with the axes of co-ordinates as 
asymptotes. But, when the visible portion 
of the diaphragm is reduced to two right 

Fic. 5. angle triangles, as in Fig. 6, the curve again 
becomes a circle. For the area of this visible portion is in that case: 
(a— x) + (ak y), 
and the illumination : 
Gr) FICHES | 
UT Le 
PAG 
hence : 

g 
Vy 
5 are oan 3) / YY 
(@— x) + (a+ y) = 2e ( Fire. 6. 
The centre of these circles is the middle of the screen’s opaque square : 
as in the transparent square, the curves of equal illumination are circular 
so long as P is within the square formed by joining the middle of the 
sides of the opaque square. 
