THE HON. J. W. STRUTT ON THE THEORY OE RESONANCE. 
97 
potential over the ends, and the solution of the modified problem is obvious. Outside 
the tube the question is the same as for a simple circular hole in an infinite plane, and 
inside the tube the same as if the tube were indefinitely long. 
Accordingly 
The correction to the length is therefore - It, that is, - R for each end, 
o 2 7 4 
ttR~ 
( 28 ) 
Helmholtz, in considering the case of an organ-pipe, arrives at a similar conclusion, — 
77 
that the correction to the length (a) is approximately ^ R. His method is very different 
from the above, and much less simple. He begins by investigating certain forms of 
mouths for which the exact solution is possible, and then, by assigning suitable values 
to arbitrary constants, identifies one of them with a true cylinder, the agreement being- 
shown to be everywhere very close. Since the curve substituted for the generating line 
of the cylinder lies entirely outside it, Helmholtz infers that the correction to the length 
thus obtained is too small. 
If, at the ends of the tube, instead of layers of matter of no density, we imagine rigid 
pistons of no sensible thickness, we shall obtain a motion whose vis viva is necessarily 
greater than that of the real motion ; for the motion with the pistons might take place 
without them consistently with continuity. Inside the tube the character of the motion 
is the same as before, but for the outside we require the solution of a fresh problem : — 
To determine the motion of an infinite fluid bounded by an infinite plane, the normal 
velocity over a circular area of the plane being a given constant, and over the rest of 
the plane zero. The potential may still be regarded as due to matter confined to the 
circle, but is no longer constant over its area ; but the density of matter at any point, 
rlfo 
being proportional to -j- or to the normal velocity, is constant. 
The vis viva of the motion 
the integration going over the area of the circle. 
The rate of total flow through the plane 
( 29 ) 
