aperture of the stop of a photographic lens 203 
the hemisphere in the neighbourhood of P, where POZ=8, 1s 
IdS.cos@. Hence the light which passes through the spherical 
cap, which contains Z% and is bounded by the line of latitude 
passing through P, is 
6 
lds i Oe imadoucos Gels? ON aS 
0 
Suppose, now, that the object is a disk of radius h, placed with 
its centre at X, on the axis of a lens system, the normal to the disk 
coinciding with the axis, and that its image is a disk of radius h, 
with its centre at X,. The arrows in Fig. 6 serve to show that, in 
the case figured, the image is inverted. Let the ray X,XY, after 
passing through the system, become the ray X,X,’ and let these 
rays make angles @, and @, with the axis; in the figure h, and 0, 
are negative. We shall suppose that all the rays from the point X, 
which pass through the system meet again in the point X,, so that 
the system is free from spherical aberration with respect to the two 
points X,, X,. Then, if every ray which leaves any point on the 
edge of the disk at X, passes through the corresponding point on 
the edge of the image disk after traversing the system, so that 
there is no coma, then h,, h., @;, 8, will satisfy the “sine condition ” 
(Onley Sid Ch fs On STM (he cecooce-conncooseneds (7) 
where wz, and », are the refractive indices of the media at X, and 
X,. When, as in our case, there is air at both ends of the system, 
fy = fo = 1 and then 
[pe 100) (oh [Da STAY (2). ek Sane bn ook onocwnaed (8) 
It is assumed that the radu h, and h, are infinitesimal. 
Tf the lenses absorb no light, all the light, which leaves the 
object disk at X, and traverses the system, reaches the image disk 
at X,. If X,X,’ be the extreme ray from X, which passes through 
the system, the amount of light which reaches the image is 
ql sin?@,.7h,; by (8), this is the same as wl sin?@,.7h2. But 
the latter is exactly the light which would pass out from the 
image disk within the angular limits defined by @,, if it were an 
actual disk with a surface of intrinsic brightness 7. The image 
may therefore be described as having the same intrinsic bright- 
ness as the object. The equality wJ sin? @,. 7h, =7I sin? 6,. 7h? 
expresses the important reciprocal result that, if the two disks are 
of equal intrinsic brightness, the light which one receives from the 
other is the same for both. 
If the light which reaches unit area of the image be denoted 
by L, we have 
Tar Tigi Go Sets Sees weet an aa Byecee is «e (9) 
and thus the speed of the lens will be proportional to sin? @,. 
