214 Dr. Norman Campbell on 



Thus we obtain 



(a + b) Ik g (E - F) + lr* (H - F)] 





0Ji(X«)Ji(x*)v- ^ 4 ) 



A. 



Hence with the aid of the preceding results a value of 

 the third elliptic integral II can be found in terms of Bessel 

 function integrals. 



The magnetic potential *>, at the point P, due to unit 

 current in the circle of radius a and centre 0,is, with 0C = - 

 e' = ±ab/(a + b)\ given by 



» = f i }irc-bCt <tt 



+ 



G 



C- 



as may be verified by calculating co as the solid angle sub- 

 tended at P by the circle. 



Thus from (15) and (19) above and the results for E and 

 F already acquired we obtain a Bessel function expression for 

 this potential. 



A large number of alternative expressions for the elliptic 

 integrals could be constructed, but it is doubtful whether any 

 of them would be found of any particular use. 



Mr. T. H. Havelock (Phil. Mag. xv. 1908) has evaluated 

 in power series some of the Bessel function integrals here 

 dealt with and applied these series to the calculation of 

 mutual and self inductance. 



XVIII. Time-Lag in the Spark Discharge. 

 By Norman Campbell, Sc.D* 



Note. — The experiments described in this paper were carried out 

 at the National Physical Laboratory under the direction of the 

 Advisory Committee for Aeronautics. Their results have been 

 communicated in a series of confidential reports to the Internal 

 Combustion Engine Sub-Committee of that committee ; per- 

 mission has now been given for the publication of any portions 

 of the work which appear of pure scientific interest. 



* Communicated by the Author. 



