146 Dr. Barton and Mr. Morton on the Criterion for 



Inserting the numerical values and arranging in powers of 

 a/*, it comes out 



1 _ R 2 1 RVV 1 RV/a 3 37 R 6 «V 

 C~4L 96 L 3 768 L 4 + 46080 L 5 * * ^ ' 



Since a= — , where Z is the length of the wire, we have the 



1 

 correcting terms expressed as a series in C- thus 



o = 4t:1 1 -^(lJ-i1)2U) + iT52oVlJ-'V' (11) 



Equations (10), (11) show that the critical capacity is 

 greater than that given by the simple theory. 



It is interesting to compare with the above another and more 

 physical way of treating the question. In a paper by one of 

 us* read before the Physical Society on January 27th of this 

 year, expressions were obtained for the equivalent resistance 

 and inductance of a wire for damped simple harmonic 

 oscillations. The method of that paper was equivalent to 

 putting for Q in equation (3) the value e (ip ~ kp)t and arranging 

 the result in the form 



0=^ + (ip-kp)R // +{ip-kpYL // . . . (12) 



The real quantities R ;/ , I/' gave the resistance and induct- 

 ance required. To apply this to the present problem, — we can 

 evidently get an approximation to the criterion sought by 



4L 

 replacing R and L in the ordinary formula C= ^ by 



modified values appropriate to the case. These may be got 

 approximately by putting, in the expressions for R" and h", 



p = and kp = —^- 

 2 J j 



The result obtained by this method agrees with (11) as far 

 as the terms there given. Let us now examine what the 

 process sketched above really amounts to as a mathematical 

 treatment of equation (4), and to what order of small quantities 

 its approximation holds good. 



First we must put ip — kp in (f> and arrange in accordance 

 with (12). Using Taylor's expansion, we have 



* Barton, Phil. Mag. vol. xlvii. pp. 433-441 (1899). 



