of Stationary States of Motion. 257 



e by es/^TT, co by j3/p and introduce c as a factor ; then we 

 obtain one-fourth of the value given elsewhere *, which is 

 apparently due to an error in Oseen's equation for R, the 

 first equation (3), where the factor 16 should be 4; this 

 error, however, does not affect our argument, and the agree- 

 ment in form, apart from the trigonometrical factor in (24), 

 verifies the substantial correctness of the expression (17) for 

 the radiation. 



When y, i. e. kajp, is small, the trigonometrical factor is 

 practically unity, but when k is comparable with p/a, i.e. of 

 the order 50,000, this is no longer true. Its presence ensures 

 the convergence of the series for all real values of /3. For a 

 surface charge the trigonometrical factor becomes 



{sin {ka\p)l(kalp)} 2 , 



but this does not suffice to secure convergence when ft 

 exceeds unity f. 



17. By the method of the last section we can also estimate the 

 error committed in the present example by the approxima- 

 tion used in obtaining (17) from (15). To obtain an estimate 

 we put vr = m'=p + a in the Bessel Function factors, and 

 z-z —a in the cosine in (15). Expanding in powers of a\p 

 and retaining only the first power in addition to the principal 

 term represented by (17), we see that the cosine term contri- 

 butes nothing to this order, whilst the Bessel Function factors 

 contribute additional terms, which in the present example 

 reduce to 



— %a$k l /3\ [l-/3 2 sin 2 0-fcos 2 0] 



x J K {k/3 sin 6)JJ{k/3 sin 0)d0. 



The coefficient of a| in the sum is 2k times the integral I3 

 introduced and evaluated elsewhere, with l-=.m — k; using 

 this value together with (23) we obtain for the additional 

 term 



+ (! + &){* 3%K{2k*)da?l. 



The form of this expression is quite similar to that of (24), 

 and the same properties may presumably be predicated of it 



* Loc. cit. p. 110. 



t For these convergence results I am indebted to Prof. G. N. Watson. 



Phil. Mag. S. 6. Vol. 36. No. 213. Sept. 1918. S 



