304 



20 



These equations show that ^ and and consequently f and ly, are functions of vo^ 

 and lüg only. From this it follows that the displacement z = ^~2~^ electron 

 in the direction of the z-axis may be expanded in a doubly infinite series of the form 



z = = 2'A^l,T3e-'^'<^•'^■ + ■^'«'•^ (54) 



where the summation is to be extended over all positive and negative entire values 

 of Tj and Tg. According to Fourier's theorem we get for Atj, tj 



Following the procedure given in § 1 we will now transform this integral in an 

 integral over c'' and y. From (52) and (53) we get for the functional determinant 

 of this transformation 



^(u;„u;,) 1 «^iCOs^+l cos c'- j 



*ûT, r = i = T-Ti(l COS -|- <T, COS 7). (56) 



^{'P.X) 4;r^ «TgCOS^ ff, cos^ + l ' ^ ' ^ 



For z = *50) 



c — ;5 M,— M, , L, cos t^' — L, cos y r/r r f2/ , ^ 



= 2 + 2^ ^ x/(/,-/2)^;f/^((7jCOsç^-ff,cos^). 



Hence, if both and are different from zero, the integral (55) assumes the form 



r»2 7r, «2; 



COS ^ — COS /)(1 + (T]^ COS ^ + cos7)e-''^-ç''~'^<''^'"^^-'^»^-'^''>s'°'^rf^ rf^, (57) 

 where r = r, + r,. 



(»2^1 '2- 



The expression (57) is equal to the sum of six terms each consisting of the product 

 of two definite integrals of the type 



constant x \ (cos e~ 4' 4' dip, (58) 



where p is equal to 0, 1 or 2. This integral will be seen to be equal to a sum of 

 Bessel coefficients of argument ro-, and of different orders, each multiplied by a 

 factor. Performing the necessary calculations, making use of (34), and contracting 

 terms by means of (35) and of 



2-(J„_l(/.)-J„ + l(/>)) = 2^Jn{p) = Snip). 



we get the final result 



