Coefficient of End- Correction. 249 



open end. The author has amplified this and assumed that 

 at the open end V is of the form 



A + B(l-*7 2 )+C(l-^ 2 ) 2 + D(l-^ 2 ) 3 . 



§ 2. As before, let us use cylindrical coordinates 'ur, z and 

 let us take the radius of the tube to be 1, and its length to 

 be L. Let us divide the whole space into two parts : first, 

 the hemisphere tsr > 0, z > 0, and ot 2 -r z 2 -<R 2 where R is 

 large ; secondly, the cylinder ■< -or < 1, — L ^ £ < 0. 



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



given by Dirichlet's condition that 1 V ^— dS is a minimum, 



where &n is an element of normal drawn outwards from the 

 region over whose surface the integral is taken. But in 

 this case we shall assume, not a given value of the current, 

 but of the potentials at z = — L and at z= +oo . 



In the region II. a solution of Laplace's equation will be 

 given by 



V = - lz + E + SflvMFoOti*) • 



r 



This satisfies the proper conditions ; for near the end z= — L 

 we have 



V=-/* + E, 



while at the boundary w=l, 



or Ji(*r)-0. 



Thus the krS must be chosen so as to satisfy this equation. 

 Then 



V* =0 =E + Sa,J (*,*r) 



r 



= A + B(l- OT 2 ) + C(l-^ 2 ) 2 + D(l-^ 2 ) 3 . 



Multiply by J (Av«r)«r and integrate from to 1. 

 Then 



«ri Jo 2 (fc) = 1 AJ (/: r 'sr)^/OT + i B(l - ot 2 ) J (lc r w) vrdw 



Jo Jo 



+ ( C{l-v^*J<fam)wdm + f ^(l-isr 2 )^*^)^ 

 Jo Jo 



Jo Jo 



1 J (k*r)(l-w s y-i v d*r = 2"- l r(v) J -4P. 

 o ** 



* Schafheitlin, 'Bessel Function?/ p, 31. 



£ 



