104 
THE HON. J. W. STRUTT ON THE THEORY OF RESONANCE. 
When 1= go , it becomes 
1-6565 E=2x -8282 R. 
Thus the correction for one end of an infinite tube is limited by 
a >-785 11 ( . (4gj 
< -8282 RJ 
When l is not infinitely great the upper limit may be calculated from (47'), the lower 
limit remaining as before; but it is only for quite small values of l that the exponential 
terms in (47') are sensible. It is to be remarked that the real value of « is least when 
1= 0, and gradually increases to its limit when 1= oo. For consider the actual motion 
for any finite value of l. The vis viva of the motion going on in any middle piece of the 
tube is greater than corresponds merely to the length. If the piece therefore be removed 
and the ends brought together, the same motion may be supposed to continue without 
violation of continuity, and the vis viva will be more diminished than corresponds to the 
length of the piece cut out. A fortiori will this be true of the reaj motion which 
would exist in the shortened tube. Thus a steadily decreases as the tube is shortened 
until when Z=0 it coincides with the lower limit tR. 
4 
In practice the outer end of a rather long tube-like neck cannot be said generally to 
end in an infinite plane, as is supposed in the above calculation. On the contrary, there 
could ordinarily be a certain flow back round the edge of tube, the effect of which must 
be sensibly to diminish u. It would be interesting to know the exact value of a for an 
infinite tube projecting into unlimited space free from obstructing bodies, the thickness 
of the cylindrical tube being regarded as vanishingly small. Helmholtz has solved 
what may be called the corresponding problem in two dimensions ; but the difficulty in 
the two cases seems to be of quite a different kind. Fortunately our ignorance on this 
point is not of much consequence for acoustical purposes, because when the necks are 
short the hypothesis of the infinite plane agrees nearly with the fact, and when the necks 
are long the correction to the length is itself of subordinate importance. 
Nearly Cylindrical Tubes of Revolution. 
The non-rotational flow of a liquid in a tube of revolution or of electricity in a similar 
solid conductor can only in a few cases be exactly determined. It may therefore be of 
service to obtain formulae fixing certain limits between which the vis viva or resistance 
must lie. First, considering the case of electricity (for greater simplicity of expression), 
let us conceive an indefinite number of infinitely thin but at the same time perfectly 
conducting planes to be introduced perpendicular to the axis. Along these the poten- 
tial is necessarily constant, and it is clear that their presence must lower the resistance 
of the conductor in question. Now at the point x (axial coordinate) let the radius of 
the conductor be y , so that its section is ry 2 . The resistance between two of the above- 
mentioned planes which are close to one another and to the point x will be in the limit 
