ELECTROSTATIC FIELD IN VACUUM TUBES 307 



charges and hence l^.v, y) gives a constant potential on the surface of each 

 grid wire if the radius of the grid wire is small compared with the spacing. 

 Wq may show by direct substitution that V{x, y) becomes equal to Vj, at 

 all points of the anode and equal to zero at all points of the cathode. On 

 the anode we have y = di which, when substituted in the expression for 

 /(•^) y), gives the logarithm of the absolute value of the ratio of conjugate 

 complex quantities, and hence 



/(:f, <f2) = 



Substituting in (1), we then readily verify that V{x, dt) = Vp. On the 

 cathode we make use of the quasi-periodicity of the t?i- function, as ex- 

 pressed by 



Uz) = -e'''^^-'^' Hz + ^r), (9) 



to prove 



f{x,-d,) = ^^, (10) 



from which it follows that V{x, — di) = 0. To show that all grid wires 

 are at the same potential, we make use of the other periodicity of the 

 i?rfunction, 



t?i(z+7r) = -t?i(z), (11) 



which shows that 



fix ± ma, y) = /(x, y), m = 0, 1, 2, • • • (12) 



It remains to prove that V actually approaches the value V, in the neigh- 

 borhood of the typical wire, which may be taken at the origin since the 

 solution repeats periodically with the wire spacing. We let 



X-}- iy = ce'^ (13) 



and assume c/a « 1. Expanding in power series in c/a, we find that the 

 first order terms are included in: 



fie cos d, c sin 0) = /n 



dx{-2Tridi/a) 



= B (14) 



t?i'(0)7rceV« 



The sign of the argument of the «?i-function in the numerator is of no conse- 

 quence since it does not affect the absolute value. Substituting back in (1), 



we then find 



Lim V{c cos d, c sin d) = Vg (15) 



c/a-»0 



The solution is thus completely established. 



