^iV OPTICAL HARMONIC ANALYZER 409 



and cos nx will in general have both negative and positive values they 

 cannot be directly represented by the transmission of light, which is 

 essentially positive. However, the addition of a constant to each 

 function will eliminate this difficulty, and merely results in a constant 

 in the measured amplitude, as shown below. 



The optical transmission of the film on which f{x) is recorded may 

 be written 



A +/(x), 



where A \s a constant large enough to make the expression positive 

 for all values of x. Similarly, the transmission of the cosine screens 

 may be written 



5n[l + Mn cos {nx - 0)], 



where ^ is a parameter denoting the position of the cosine screen along 

 the X-axis, Mn is a constant somewhat less than unity, known as the 

 modulation of the record, and J5„ is a constant which is seen to be 

 the average optical transmission of the screen. 



If one or both of these records is cf the variable density type, the 

 total transmission when they are superimposed will be 



T = { B,,IA +/(x)][l + Mn cos {nx - d)yx 



= I ABndx + I BrJ{x)dx + | ABnMn COS {nx - d)dx 

 Jo Jo Jo 



BnMJ{x) COS l{nx - (pn) - {0 - 0„)](/x, 





 = 2wBn{A + Co) + TrB„AfnCn COS {d - 0„). (8) 



To obtain a„, we take the difference in T for 6 = and = w, which is 

 seen to be 



lirBnMnCn COS 0„ = 27r5,J/„a„. (9a) 



iDtam On, we make ^ = -^ and t) = 

 difference in T 



Similarly, to obtain hn, we make ^ = -^ and 6 = -■- , giving for the 



lirBnMnCn slu 0„ = lirBnMnbn- (9b) 



To obtain c„ and 0„ note that the maximum value of T occurs at 

 6 = (j)n and the minimum at = 0„ + tt, which serves to determine </>„. 

 The difference between the maximum and minimum values of T is 



lirBnMnCn. (10) 



