THE HON. J. W. STRUTT ON THE THEORY OF RESONANCE. 
99 
and so from (29) 
2 vis viva 8 
(rate of flow) 2 3w 2 R 
This is for the space outside one end. For the whole tube and both ends 
2 vis viva - L J_ 16 
(rate of flow) 2 wR 2 1 3w 2 R 
(32) 
(oo) 
Whatever, then, may be the ratio of L : R, the electrical resistance to the passage in 
question or - is limited by 
1 JL 1 
c > 7tR 2 + 2R 
L 16 
wR 2 * 3 t 2 R 
(34) 
In practical application it is sometimes convenient to use the quantity a or correction 
to the length. In terms of a (34) becomes 
^ T1 i 
ct q -L i 
< 37T lv ’. 
or in decimals, 
«>(l-571R=2x •785R)1 
1 ; (34') 
C4 < (T697R=2 x , 849R)j v 7 
The corrections for both ends is the thing here denoted by a. Of course for one end 
it is only necessary to take the half*. 
I do not suppose that any experiments hitherto made with organ-pipes could discri- 
minate with certainty between the two values of « in (34'). If we adopt the mean pro- 
visionally, we may be sure that we are not wrong by so much as ‘032 R for each end. 
Our upper limit to the value of a expressed in (34') was found by considering the 
hypothetical case of a uniform velocity over the section of the mouth, and we fully deter- 
mined the non-rotational motion both for the inside and for the outside of the tube. 
Of course the velocity is not really uniform at the mouth ; it is, indeed, infinite at the 
edge. If we could solve the problem for the inside and outside when the velocity 
(normal) at the mouth is of the form a-\-br 2 , we should with a suitable value of b : a get 
a much better approximation to the true vis viva. The problem for the outside may be 
solved, but for the inside it seems far from easy. It is possible, however, that we may 
* Though not immediately connected with our present subject, it may he worth notice that if at the centre 
of the tube, or anywhere else, the velocity be constrained (by a piston) to be constant across the section, as it 
would approximately be if the tube were very long, without a piston, the limiting inequalities (34) still hold 
good. For large values of L the two cases do not sensibly differ, but for small values of L compared to R the 
true solution of the original problem tends to coincide with the lower limit, and of the modified (central piston) 
problem with the higher. 
