204 Dr Searle, The determination of the effective 
The system cannot be free from spherical aberration and from | 
coma for all pairs of conjugate points, and it thus becomes necessary 
to accept a lower standard and to work on that plane of approxima- 
tion where the angles are treated as being so small that they may 
be used instead of their sines or tangents. We then have 
Le = TIO Sion snc eee (10) 
We can now apply these results to determine the effect of the 
position of the object upon the speed of the lens. 
Fig. 7. 
In Fig. 7, let S be the actual stop, 7, its image formed by the 
front lens Z (Fig. 2) and 7, its image formed by the back lens M; 
the lenses themselves are not shown in Fig. 7. Let O,Q, = q, and 
O.Q.=q, be the radi of the two images of the opening of the 
stop. ‘Then q, 1s identical with 4a, where a is the quantity used in 
§§ 2,4. IfX, is a point on the axis and_X, is its image, one of the 
extreme rays leaving X, and reaching X, is the incident ray X,Q; 
which gives rise to the emergent ray Q,X.. Let these make angles 
6,, 0, with the axis, where 0,=¢q,/0,X, and @,=4,/0,X.. Let be 
the angle subtended by Q,0O, at the focus F,, where the plate is 
placed when a distant scene is to be photographed. Then 
b= q,/0.F,. We see, at once, that, when X, is beyond Ff, as 
is the case when a “real” object is to be photographed, the 
angle 6, is less than the angle ¢ and that, in consequence, the 
speed of the lens is less when the object is at a finite distance than 
when it is at an infinite distance from the lens, for, by (10), the 
speeds are proportional to 6 and ¢*. 
If L,, be the light received per unit area by the image when 
the object surface of standard intrinsic brightness J is at an 
infinite distance, we have 
L = 710.2 = w1q?/0,X 22, 
L,, = wg? = 119: O.Fe, 
and thus 
L, (oAl= (nt nas) if (1 fi EXE 
aan a he aah 
