THE CALCULATION OF MODULATION PRODUCTS 233 



Z„. = 1 = dz. (14) 



By differcMitiatint;- the expression 2"'-3 V(l - s2)(l — ^V), we may 

 easily derive the useful recurrence formula: 



{m - 2)(1 + k')Z,n-2 - (m - 3)Z„.-4 

 (m - l)yfe2 



Z„j — 777^ 1 \ 7 2 V^-^/ 



We may now calculate the value of Z,„ for even values of m in terms of 

 Zo and Zs. Zo is a complete elliptic integral of the first kind which we 

 shall designate as usual by K; i.e., 



Zn^K^ r "^^ f' \ 1 - '^^ sin2 0C/0. (16) 



Jo \(1 - C^)(l - Fc.'^) ,/o 



Furthermore from the identity: 



1 |1 _ fe2^2 



, (17) 



\'(1 - 22)(1 - F, 



we have 



:2) ^4\'(1 - s'^)(l - A;^c-) ^' 1 



Z2 = 7^(i^-£), (18) 



where £ is a complete elliptic integral of the second kind defined by 

 E= f yj \''J'f dz = p ^l- k-'sm'edd. (19) 



Now making use of (15), we calculate Z4 in terms of Z2 and Zo and 



get finally: 



^ (2 + k'')K - 2(1 + P)£ .^^^ 



We can then evaluate (13) in terms of K and E. The result is 



^11 = ^ C(l + ^'^)£ - (1 - ^')^]- (21) 



The process of evaluating the other coefficients is quite similar. 

 Results are listed in Table I. 



Convenient tables of K and E may be found in Peirce's Short Table 

 of Integrals (page 121), Byerly's Integral Calculus, and the Jahnke 

 und Emde tables. For a very extensive set of tables, see Legendre's 



