Lengtli of Terminated Rods in Electrical Problems. 9 

 Also 



(j) i { k l a)=--. 1 -, ( f ) '(I,^) = ^Ml. . (36) 

 T ' k L a log k Y a T * log k x a v 7 



Using these, we find from (16) and (24) 



b 1 H, Hf 



.^ 7:373 T 27 7.7, N T" /_ it ■ _. li \T .1. L\ I / /. /. v + a \ 1 



log 2 6/a~W Ji 2 (^) T k^biogbja . Y vV>) Ji(M) _r 4A: 1 Y 2 (/: 1 Z>) 



as the expression for the capacity which is to be made a 

 minimum. Comparing the terms in H : 2 , we see that the 

 two last, corresponding to the negative side, vanish in com- 

 parison with the other in virtue of the large denominator 

 log 2 k x a. Hence approximately 



xt _ _ 2Y (kib) 



1- ktblogbja.J^kyb)' ' ' ' ' ^ } 

 and (37) becomes 



I b «, 1 b 1 



+ 



2 log b/a T log* b/a ¥¥ J } \kb) log 2 b/a k^ J^kji) 



. . . (39) 



when made a minimum by variation of Hj. Thus the effect 

 of the correction depending on tlie introduction of Hj is 

 simply to wipe out the initial term of the series which 

 represents the first approximation to the correction. 



After this it may be expected that the remaining terms of 

 the first approximation to the correction will also disappear. 

 On examination this conjecture will be found to be verified. 

 Under each value of k in (Hi) only that part ot B is important 

 for which h has the particular value which is nearly equal to 

 /•. Thus each new H annuls the corresponding member of 

 the series in (39), so that the continuation of the process 

 leaves us with the first term of (39) isolated. The inference 

 is that the correction to the capacity vanishes in comparison 

 with b-r- log 2 b/a, or that 81 vanishes in comparison with 

 b-r- log b\a. It would seem that 81 is of the order b-±- log 2 b\a, 

 but it would not be easy to find the numerical coefficient by 

 the present method. 



In any case the correction 67 to the length of the rod 

 vanishes in the electrostatical problem when the radius of 

 the rod is diminished without limit — a conclusion which I 

 extend to the vibrational problem specified in the earlier 

 portion of this paper. 



