The mathematical determination of Q'x/Qx for vignette free pencils of rays does not 
lead to a direct solution. 
For the mathematical analysis and adequate determination of the loss of illumination, 
an examination of the width of the pencil along the meridian section is sufficient. 
Designating as a a and b a the semi-axes of the elliptical discs of light given by a vignette 
free diaphragm (Figure 3), the relations a a /a 0 very closely approach the values of 
Qoc/Qx since the bx deviate only slightly from the b Q . 
In general, if a a is considerably larger than a Q , then b a will be greater than b Q . This 
deviation tendency may also be seen by a comparison of the values of Q' a / Q 0 for 
a = 45° from Figure 4 with the corresponding values of a a /a 0 as given in the following 
table. The table gives the maximum calculated values of a a /a 0 for several objectives 
with vignette free diaphragms at different incidence angles ; the values are taken from 
the patent descriptions. 
Nr. 
Objective 
Figure 
A a / A o f or 
348 
458 
6C g 
1 
Old wide angle type 
6 
1.00 
0.84 
0.67 
2 
Liar 6 by Russinow-Kosyrev 
7 
1.00 
1.01 
1.03 
1.04 
3 
Steinheil objective 
English patent 21211/1901 
1 
1.00 
1.06 
1.24 
4 
Russar U.S. patent 2516724 
French patent 935617 
8 
1.00 
1.03 
1.27 
1.68 
5 
Aviogon by Bertele 
5 
1.00 
1.20 
1.40 
The table shows that the ordinary wide angle objectives formerly used, even without 
vignetting, have an illumination loss worse than that given by cos 4 a . The objective 
Fig. 7 Section through “Liar 6” objective by Russinov-Kosyrev. (From: Untersuchungen über 
Aerovermessungen und Photogrammetrie, Moskau 1939.) The light loss of this objective 
corresponds nearly to cos 4 a.