THE HON. J. W. STETTTT ON THE THEOEY OF EESONANCE. 
83 
may say that the mass to be moved is the air in the neighbourhood of the opening, and 
that the air in the interior acts merely as a spring in virtue of its resistance to compres- 
sion. Of course this is only true under the limitation specified, that the diameter of the 
reservoir is small compared to the quarter wave-length. Whether this condition is 
fulfilled in the case of any particular resonator is easily seen, a posteriorly by calcu- 
lating the value of X from (6), or by determining it experimentally. 
Several Openings. 
When there are two or more passages connecting the interior of the resonator with 
the external air, we may proceed in much the same way, except that the equation of 
energy by itself is no longer sufficient. For simplicity of expression the case of two 
passages will be convenient, but the same method is applicable to any number. Let 
X,, X 2 be the total flow through the two necks, e n c 2 constants depending on the form 
of the necks corresponding to the constant c in formula (6); then T, the vis viva , is 
given by 
T: 
x? x; 
c, C a 
the necks being supposed to be sufficiently far removed from one another not to inter-' 
fere (in a sense that will be obvious). Further, 
V= Potential Energy — | h 0 a 2 - ^ - • 
Applying Lagrange’s general dynamical equation, 
we obtain 
dT dV 
d\p c/vf ’ 
as the equations to the motion. 
By subtraction, 
or, on integration, 
l+Jx.+x^o, 
X 
+ s(X 1 +X 2 )=0 
0 
0, 
( 7 ) 
(S) 
Equation (8) shows that the motions of the air in the two necks have the same pe- 
riod and are at any moment in the same phase of vibration. Indeed there is no essen- 
tial distinction between the case of one neck and that of several, as the passage from 
one to the other may be made continuously without the failure of the investigation. 
