THE CALCULATION OF MODULATION PRODUCTS 241 



When three frequencies are applied, a triple Fourier series is required, 

 and in the general case of n frequencies, a Fourier series in n variables 

 would be used. The work becomes more complicated as the number of 

 frequencies increases, but there is no theoretical limitation. 



In conclusion the writer wishes to express his appreciation of the 

 valuable advice of Messrs. T. C. Fry and L. A. MacCoil on the tech- 

 nical features of the paper. 



Appendix 



The hypergeometric function F(a, /3; 7; z) may be defined by the 

 power series: 



F(a, /3; ,; =) = 1 + if = + "j;^l)M + l) . + . . . . 



1!7 217(7 + 1) 



When any one of the three quantities a, ^, 7 is increased or decreased 

 by unity a new hypergeometric function is formed which is said to be 

 contiguous to the first. Gauss listed fifteen linear relations which 

 connect F(a, /S; 7; 2) with pairs of its contiguous functions. In deriv- 

 ing the recurrence formula for yl„,„ we require difference relations 

 between functions which are not contiguous, but the required relations 

 may be obtained from those listed by Gauss by a process of substitution 

 and elimination. 



We shall find it convenient to designate F(a, /3; 7; s) by F, F(a -f 1, 

 /3; 7; s) by F„+, F(a + 1, ^; y — l; z) by Fa+y-, etc., and to let 



m -f w — 3 ^ n — m — I 

 « = 2 ' ^ ^ 2 ' "Y = ^^ " = ^ ■ 



In this notation. Equation (27) becomes: 



(-)— +-r / "' + '' -' u.p 



A — 



2.r(„ + i)r ( "' - ^' + ^ ) 



«+7+' 



The corresponding expressions for A,n-\, n-i and ^,„_2, «-2 are by direct 

 substitution: 



m-\-n , . 



Am-\, n-1 — ■ , TT F, 



\n)V (^ 



Am— 2, n-2 = ; j Z" Fa-y — 



