96 
THE HOIST. J. TV. STRUTT OX THE THEORY OE RESONANCE. 
independently of each other, and the value of c for the pair is the simple sum of the 
separate values, as may be seen either from the law of multiple arcs by considering c 
as the electric conductivity between the outside and inside of the reservoir, or from the 
interpretation of M in (23). The first method applies to any kind of openings with or 
without necks. As the two circles (which for precision of statement we may suppose 
equal) approach one another, the value of c diminishes steadily until they touch. The 
change in the character of the motion may be best followed by considering the plane 
of symmetry which bisects at right angles the line joining the two centres, and which 
may be regarded as a rigid plane precluding normal motion. Fixing our attention on 
half the motion only, we recognize the plane as an obstacle continually advancing, 
and at each step more and more obstructing the passage of fluid through the circular 
opening. After the circles come into contact this process cannot be carried further ; but 
we may infer that, as they amalgamate and shape themselves into a single circle (the 
total area remaining all the while constant), the value of c still continues to diminish 
till it approaches its minimum value, which is less than at the commencement in the 
ratio of \/2 : 2 or 1 : V2. There are very few forms of opening indeed for which the 
exact calculation of M or c can be effected. We must for the present be content with 
the formula (26) as applying to nearly circular openings, and with the knowledge that 
the more elongated or broken up the opening, the greater is c compared to a. In the 
case of similar orifices or systems of orifices c varies as the linear dimension. 
Fig. 4. 
Cylindrical ICeclcs. 
Most resonators used in practice have necks of greater or less length, and even where 
there is nothing that would be called a neck, the thickness of the side of the reservoir 
could not always be neglected. For simplicity we shall take the case of circular cylinders 
whose inner ends lie on an approximately plane part of the side of the vessel, and whose 
outer ends are also supposed to lie in an infinite 
plane, or at least a plane whose dimensions are 
considerable compared to the diameter of the 
cylinder. Even under this form the problem does 
not seem capable of exact solution ; but we shall 
be able to fix two slightly differing quantities 
between which the true value of c must lie, and 
which determine it with an accuracy more than 
sufficient for acoustical purposes. The object is to 
find the vis viva in terms of the rate of flow. Now, according to the principle stated at 
the beginning of Part II., we shall obtain too small a vis viva if at the ends A and B of 
the tube we imagine infinitely thin laminae of fluid of infinitely small density. We may 
be led still more distinctly perhaps to the same result by supposing, in the electrical 
analogue, thin disks of perfectly conducting matter at the ends of the tube, whereby the 
effective resistance must plainly be lessened. The action of the disks is to produce uniform 
