138 Prof. P. J. Daniell on the 



the author's paper in the American Journal o£ Mathematics 

 and Rayleigh's e Theory of Sound,' Appendix A, may be 

 read. 



§ 2. Let us use cylindrical coordinates -nr, z, and let us 

 take the radius of the tube to be 1, which will not affect the 

 general nature of the problem, and its length to be L. 



Let us divide the whole space into two parts : firstly, the 

 hemisphere «r > 0, z > 0, and ot 2 -f-£ 2 < R 2 where R is large ; 

 secondly, the cylinder < -nr < 1, — L-<^<0. 



Let V be the potential at any point, then a solution is 



given by Dirichlet's condition that 1 Y^—d$ is a minimum, 



where ~dn is an element of normal drawn outwards from the 

 region over whose surface the integral is taken. If the 

 current is given and equals tt, which can be assumed without 

 loss of generality, this minimum value will come out as 

 7r(L + D) and will be proportional to the total resistance. 

 The coefficient of end-correction will then be this D, or less 

 than this D *. 



In the region I. a solution f of Laplace's equation is given 

 by 



V= n- fa J (for)rf£ Cf(p)J (kp)pdp 9 

 Jo Jo 



and this satisfies the conditions, V = when «r and z are 

 infinite, and 



1 )_.-*->■ 



But there is a boundary of the conducting material where 

 z = and «r >■ 1, or f(vr) = when «r >- 1. 

 So that 



d 



'0 



Or 

 and 



V = f °° e-^J (k^)dk( 1 f(p)J (kp)pdp. 



Jo Jo 



(V), =0 = r Ukm)dk^/(p)J^kp)pdp, 



Further, at the other boundary -sr 2 + z 2 = R 2 , V is proportional 



to =r and ^- to ™ , and the integral I V ^- dS over this 

 R dn R 2 ' & J on 



* Eayleigh, ' Theory of Sound,' vol. i. Appendix A. 

 t Lamb, ' Hydrodynamics,' p. 129. 



