THE HON. J. W. STEUTT ON THE THEOEY OF RESONANCE. 
91 
which agree nearly with the results of Helmholtz. 
defined by the equation 
so that 
Byfc — Cot 
cot #a=tan Jcl by (15), 
or 
k{l-\-a) — (2mfi-l) ^ >' 
In his notation a quantity a is used 
a may accordingly be considered as the correction to the length of the tube (measured, 
however, in our method only on the negative side of the origin), and will be given by 
cot ku — — 
kQ 
The value of c will be investigated in Part II. 
The original theory of open pipes makes the pressure absolutely constant at the 
mouth, which amounts to neglecting the inertia of the air outside. Thus, if the tube 
itself were full of air, and the external space of hydrogen, the correction to the length 
of the pipe might be neglected. The first investigation, in which no escape of energy is 
admitted, would apply if the pipe and a space round its mouth, large compared to the 
diameter, but small compared to the wave-length, were occupied by air in an atmosphere 
otherwise composed of incomparably lighter gas. These remarks are made by way of 
explanation, but for a complete discussion of the motion as determined by (13) and (17), 
I must refer to the paper of Helmholtz. 
Long Tube in connexion with a Reservoir. 
It may sometimes happen that the length of a neck is too large compared to the 
quarter wave-length to allow the neglect of the compressibility of the air inside. A 
cylindrical neck may then be treated in the same way as the organ- pipe. The potential 
ot plane waves inside the neck may, by what has been proved, be put into the form 
4 = A / sin k(x—u) cos 2 mt ; 
if we neglect the escape of energy, which will not affect the pitch of the resonant note, 
= — 2 . 7 mA! sin k(x— a) sin 2 smtf, 
- 7 —Jc A- cos k(x—ci) cos 2 \rnt. 
where a is the correction for the outside end. 
The rate of flow out of S=Q 
dx 
Total flow 
= Q f 
dA 
dx 
dt=ki AQ cos/l'L 
sin 2 Tint 
27 m 
