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IX. The Coefficient of End- Correction.— -Part I. By P. J. 

 Daniell, B.A., Assistant Professor in Applied Mathe- 

 matics, The Rice Institute, Houston, Texas *. 



§ 1. TF an electrical current passes through a long cylindrical 

 X tube of conducting material, and then out into a 

 large hemispherical volume of the same, the total resistance 

 is proportional to the total length of the tube plus a certain 

 multiple of the radius. This multiple is the coefficient of 

 end-correction which we require to find. Rayleigh, in his 

 Theory of Sound,' found first that *785 < this coefficient 

 k < *845. In the appendix he showed further that k < *8242, 

 and he supposed that its true value did not differ greatly from 

 this. The solution depends on assigning some form to the 

 current-flow across the open end stated as a function of the 

 distance ot from the central axis. Rayleigh assumes the axial 

 velocity to be of the form 1 + yu/sr 2 + /i/^ 4 , taking the radius 

 of the tube to be 1. In a paper being published in the 

 American Journal of Mathematics the author has assumed a 

 general law of the form 



R 



where k r is the rth root of J 1 (l')=0. R was taken as 7, and 

 then the coefficient required was found to be less than *8222. 

 The convergence was slow, and in the work the coefficients, 

 u r , were found to be roughly the same as if the axial velocity 

 were of a form (1 — ot 2 ) -13 . 



In the corresponding two-dimensional case the disconti- 

 nuity at the edge of the opening is just of this type. Again, 

 Rayleigh found that the -cr 4 term was of greater importance 

 than the -cr 2 . All these considerations have led the author to 

 assume a form 



A + B(l-^ 2 ) + (Xl-^ 2 )- 



1 3 



for the axial velocity, or current. This forms the substance 

 of this paper, and it is found 



(i.) That B is small and its effect on k almost negligible; 

 (ii.) That if the total current is it and the radius 1 the 



axial velocity is very nearly J- + i(l — ^ 2 ) -1/3 ; 

 (iii.) That A-<*821G8 and is probably extremely close to 

 this value. 



For a more detailed explanation of the methods employed 

 • Communicated by the Author. 



