Length of Terminated Rods in Electrical Problems. 7 



(16) and (24). The first approximation is obtained by 

 omitting all the quantities H, so that the B's vanish also. 

 The additional capacity, derived entirely from (16), is then 

 \b 2 *Z JcA 2 J l 2 (kb), or on introduction of the value of A, 



b >? Jp~(^ a ) ^ 9 kx 



log^/a AWJi 8 ^)' ' "•• ■ ^ D) 



the summation extending to all the roots of J (kb) = 0. Or 

 if we express the result in terms of the correction 81 to the 

 length (for one end), we have 



6L ~ log b/a*k*b* J {'(kby • • • • W 



as the first approximation to 81 and an overestimate. 



The series in (26) converges sufficiently. J 2 (ka) is less 

 than unity. The mth root of J (a') = is «r = (m — j)7r 

 approximately, and J 1 2 (x) = 2/7rx, so that when m is great 



^ 3 J 1 2 (^) = 7r(4m-l) :4 * ' ' " " ( 2? ) 



The values of the reciprocals of x* 3 2 {x) for the earlier roots 

 can be calculated from the tables * and for the higher roots 

 from (27). I find 



on. 



X. 



±J». 



ar-S-^V). 



1 



2-4048 



5-5201 



86537 



11-7915 



14-9309 



•51915 

 •34027 

 •27145 

 •23245 

 ■20655 



•2668 

 •0513 

 •0209 

 •0113 

 •0070 



2 



3 



4 



5 





The next five values are '0048, -0035, '0026, '0021, '0017. 

 Thus for any value of a the series in (26) is 



•2668 J 2 (2-405 a/6) + '0513 J 2 (5'520 a/b) + ... ; (2S) 



it can be calculated without difficulty when a/b is given. 

 When a/b is very small, the J's in (28) may be omitted, and 

 we have simply to sum the numbers in the fourth column of 

 the table and its continuation. The first ten roots give 

 •3720. The remainder I estimate at '015, making in all 

 *387. Thus in this case 



•774/) 



s *=ii^ < 29 > 



* Gray & Mathews, Bessel's Functions, pp. 244, 247. 



