This is all the more astonishing, because very simple experiments and considerations 
will show that the cosine fourth law is exact only for a diaphragm in the object space 
in front of the lens and in no case for a diaphragm between the lens elements. It is 
obvious that in the latter case the cosine fourth must be multiplied with a function 
which depends on the construction of the objective. This function can be determined 
either experimentally or mathematically. 
In spite of the fact that the simplest experimental determination can be made in the 
same way as the measurement of the vignetting, it would be incorrect to call it the 
“vignetting function”. The actual vignetting is only one component of the function; 
the second depends upon the property of the vignette-free passage of a pencil of rays 
through the front half of the objective. While the variable vignetting factor depending 
upon the angle of incidence may only be equal to, or less than, one, the second com 
ponent determined by the lens construction can in a favourable case be considerably 
higher than one. This leads to the conclusion that it is possible to design a distortion 
free lens in which the second factor is greater than one over the entire image area. 
The experimental determination of the function for the illumination loss is known. 
Points of light situated in the picture plane are projected on a plane in front of the 
objective and perpendicular to the lens axis. Depending upon the lens construction, 
the diaphragm, and the incident angle a, each point will be imaged on this plane as a 
disc having an area Q' a . The relation of Q ; a to Q 0 in the picture center where a = 0 
shows whether the illumination loss of an objective is better, equal to, or worse than 
that given by the cosine fourth law, depending upon whether Q' a / Q 0 is greater, equal 
to, or less than one. The percentage of light loss from the picture center is calculated 
from: 
Fig. 2 Determination of light loss for objectives Fig. 3 Determination of light loss for objectives 
with front diaphragm. Under vignette free con- with internal diaphragm. The size of the image 
ditions the image discs are circles of equal size. discs depends upon the construction of the lens.