154 Dr. T. J. I'a. Bromwich on 



It is now easy to show that the solutions (3*5) are 

 equivalent to the Cartesian solutions given by Prof. A. E. 

 H. Love *. For these solutions are known to give a radial 

 component of electric force proportional to 



-* + l)*(-i*)-{^},. . (3-51) 



where co n is a solid harmonic of order n ; while the radial 

 magnetic force is zero. 



Thus, on comparison with (3*5) we see that Love's radial 

 components of force agree with those found above ; and 

 hence, by the general theorem proved in § 2, it follows that 

 all the other components of the fields must also agree. 



A direct verification of the agreement is not difficult; 

 but the details of the work are somewhat tedious. Some 

 special examples will be found in §§ 4, 6 below. 



§ 4. The field of an electro?! oscillating in a known manner 

 along the axis of z, with the origin as its mean position. 



We take this problem as an example of the general 

 formulae of § 3 ; then we write 



n=l, Y 1 = cos6', ..... (4-1) 



and also yu = 1, K = 1, so that c x = c. 



Thus equation (3*49) gives 



where the solution is restricted to divergent waves. 

 On substituting from (4*1) and (4'2j in (3*5) we find 



X = -I (/*+ rf) cos 0, cu = 0, 



r° 



Z =0, 



e/3 = 0, >(4-3) 



<"y = J(/'+tf'>"0J 



On considering the character of (X, Y, Z) in (4*3) for 

 small values of r the field is readily recognized as due to 

 an electric doublet of strength j(ct) ; and this may be 

 interpreted as due to an oscillating electron in the usual 

 way. 



* Phil. Trans. Roy. Soc. A, vol. 197. p. 10 (1901) ; or Proc. Lond. 

 Math. Soc. ser. 2, vol. ii. p. 92 (1904). 



