408 



BELL SYSTEM TECHNICAL JOURNAL 



function is represented by gradations in the density of the film such 

 that the Hght transmission at any point is proportional to the function, 

 the density being uniform in a direction perpendicular to the axis. 

 Such a record is shown schematically in the lower part of Fig. 1. With 

 either type of representation of a function g{x), it will be seen that the 

 amount of light transmitted through a narrow vertical strip of width 

 dx is proportional to g{x)dx. If two or more such records are super- 

 imposed, the light transmitted through all cf them will be proportional 

 to the product of the recorded functions, provided not more than one 

 of the records is of the variable area type. 



Fig. 1 — Representation oi f{x) and cos nx on film. 



The determination of a„ and 6„ is now very straight-forward. 

 Suppose we have fix) recorded on one film and cos nx on another. 

 For illustration we will assume that f{x) is recorded in variable area 

 and cos nx in variable density, as shown in Fig. 1, although the only 

 necessary requirement is that they shall not both be variable area. 

 If the two films are superimposed, the amount of light transmitted 

 through both of them between the limits zero and lir is just the first 

 integral in (5) and hence proportional to a„. Similarly, if the cosine 

 screen is moved a quarter wave-length along the axis it becomes sin nx 

 and we have at once 6„. If the cosine screen is moved a whole wave- 

 length along the axis, the transmitted light will go through a maximum. 

 It will be shown below that this maximum value is proportional to c„, 

 and the position of the cosine screen at which it occurs is 0„. 



One more matter needs to be considered before we write down the 

 expressions which describe the operation of the analyzer. Since f{x) 



