THE CALCULATION OF MODULATION PRODUCTS 229 



convenient to represent the amplitude ratio Q/P by k, and without 

 loss of generality to take 



P > and < yfe < 1 . (2) 



The problem we now consider is the resolution of the output wave into 

 sinusoidal waves, a complete solution requiring the determination of 

 the frequencies present, their amplitudes, and their phase relations. 



The method of solution used employs the auxiliary function of two 

 independent variables /(.v, v) defined by 



J{x, 3') = P(cos X + k cos y), cos .v + ^ cos j S; 0, 1 ,^. 



= 0, cos X + k cos J < 0. J 



It is clear that the function /(.v, v) may be represented by a surface 

 which does not pass below the .vv-plane and which coincides with the 

 x^'-plane throughout certain regions which are bounded by the multi- 

 branched curve, 



cos X + k cos 3' = 0. (4) 



If either .r or 3' is increased or decreased by any multiple of 2-k, the 

 . value of /(.v, v) is unchanged. Hence /(:x:, y) is a periodic function of 

 X and 3', and if its value is known for every point in the rectangle 

 bounded by 3' = ± tt, .v = ± tt say, the value of the function may be 

 determined for any point in the entire .T3'-plane. 



From the above considerations we are led to investigate the expan- 

 sion of /(.r, 3') in a double Fourier series in x and y. We may readily 

 verify that the function satisfies any one of several sets of sufficient 

 conditions ^ to make such an expansion valid. We may write the 

 expansion thus: 



f{x, 3O = Z £ [^±mn COS {mx ± «30 + B±,nn siu {fux ± ;oO]. (5) 



with the summation to be extended over both the upper and lower of 

 the ambiguous signs except when m or n is zero, in which case one value 

 only is taken (it is immaterial which one) ; when m and n are both zero, 

 we divide the coefficient ^00 by two in order that all the yl -coefficients 

 may be expressed by the same formula. Determining the coefficients 

 by the usual method of multiplying both sides of (5) by the factor the 

 coefficient of which is to be found and integrating both sides throughout 

 the rectangle bounded by .r = ±7r, 3' = ±7r, we obtain: 

 1 Hobson, "Theory of Functions of a Real \'ariable," \'ol. 2, p. 710. 



