282 Mr. S. Butterworth on a Method for 



from which 



Using the value of M given by (1), and remembering that 

 £ 2 = -. 2 , we obtain from (7) 



jdz j*M <fe= g {(1 -F)K- (1-2F)E}. J 



Hence the self-induction sought is 



L = 2n 2 (\lz Mdz, 

 */o o 



= lp 2 {(l-* 2 )K-(l-2F)E-P}. 



This holds when the diameter is unity. If the diameter 



4a 2 

 4a 2 + b 2 



Aa 2 

 is 2a and the length b, put z = b/'2a } so that ^=7-2 — 72 > and 



(by dimensions) multiply by (2a) 3 . Then 



L=¥^{(1-^)K-(1-2F)E-F}. . (8) 



This expression was first given by Lorenz *. Proofs have 

 also been supplied by Coffin f and by Russell J. The author 

 believes the above proof to be new. 



6. By treating equation (8) in the same way as (1) was 

 treated, various series can be obtained for L. 



Thus, 



32 

 L + -Q- irn 2 a z 



satisfies the equation 



* Wied. Ann. vii. p. 161 (1879). 



t Bull. Bureau of Standards, ii. p. 123 (1906). 



% Phil. Mag-, xiii. p. 445 (1907). 



