OF AIR OBSERVED IR KUNDT’S TUBES. 
21 
We have seen that the width of the direct current along the wall is '423 y lt and 
that of the return current (measured up to the plane of symmetry) is *577 y lf so that 
the direct current is distinctly narrower than the return current. This will be still 
more the case in a tube of circular section. The point under consideration depends 
only upon a complementary function analogous to (8fi), and is so simple that it may 
be worth while to investigate it. 
The equation for \Jj is 
(97). 
but if we suppose that the radius of the tube is small in comparison with Jc 2 may be 
omitted. The general solution is 
xjj= {A+E>r 2 +BV 3 log r+Cr 4 } sin 2kx .(98), 
so that 
u=\ {2B + B'(2 log?’d- l)-f-4GV 2 } sin 2for, 
whence B'=:0, by the condition at r„=0. Again 
l cbjr__ _2j c $ A r i_|_Br+Cr 3 } cos 2 Tex, 
r ax 
whence A = 0. 
We may take therefore 
(99). 
If v=0, when r— R, B+CI1. 2 =0, and 
n = 2C(2r 3 —R 2 ) sin 2 kx 
( 100 ). 
Thus u vanishes, when 
The direct current is thus limited to an annulus of thickness '293 It, the return 
current occupying the whole interior, and having therefore a diameter of 
2X707 R= 1*414 R. 
