THE HON. J. W. ST11UTT ON THE THEOEY OF RESONANCE. 
81 
this paper. Moreover the motion in the passage and its neighbourhood will not differ 
sensibly from that of an incompressible fluid, and its energy will depend only on the rate 
of total flow through the opening. A quarter of a period later this energy of motion 
will be completely converted into the potential energy of the compressed or rarefied air 
inside the reservoir. So soon as the mathematical expressions for the potential and 
kinetic energies are known, the determination of the period of vibration or resonant 
note of the air-space presents no difficulty. 
The motion of an incompressible frictionless fluid which has been once at rest is sub- 
ject to the same formal laws as those which regulate the flow of heat or electricity 
through uniform conductors, and depends on the properties of the potential function, to 
which so much attention has of late years been given. In consequence of this analogy 
many of the results obtained in this paper are of as much interest in the theory of elec- 
tricity as in acoustics, while, on the other hand, known modes of expression in the former 
subject will save circumlocution in stating some of the results of the present problem. 
Let h 0 be the density, and <p the velocity-potential of the fluid motion through an 
opening. The kinetic energy or vis viva 
the integration extending over the volume of the fluid considered 
s, 
■ o o 
by Green’s theorem. 
d<p 
Over the rigid boundary of the opening or passage, =0, so that if the portion of 
fluid considered be bounded by two equipotential surfaces, (p l and <p.,, one on each side of 
the opening, 
vis viva—\ Ji 0 (<p l — <p 2 ) 1 i hfa — <p 2 )X, 
if X denote the rate of total flow through the opening. 
At a sufficient distance on either side <p becomes constant, and the rate of total flow 
is proportional to the difference of its values on the two sides. We may therefore put 
X 
c 
where c is a linear quantity depending on the size and shape of the opening, and repre- 
senting in the electrical interpretation the reciprocal of the resistance to the passage of 
electricity through the space in question, the specific resistance of the conducting matter 
being taken for unity. The same thing may be otherwise expressed by saying that c is 
the side of a cube, whose resistance between opposite faces is the same as that of the 
opening. 
