78 
THE II OX. J. W. STRUTT OX THE THEORY OE RESOXAKCE. 
note, and a the velocity of sound, 
fieri 
Helmholtz’s theory is also applicable when there are more openings than one in the 
side of the vessel. 
In the present paper I have attempted to give the theory of vibrations of this sort 
in a more general form. The extension to the case where the communication with 
the external air is no longer by a mere hole in the side, but by a neck of greater or 
less length, is important, not only because resonators with necks are frequently used in 
practice, but also by reason of the fact that the theory itself is applicable within wider 
limits. The mathematical reasoning is very different from that of Helmholtz, at least 
in form, and will I hope be found easier. In order to assist those who may wish only 
for clear general ideas on the subject, I have broken up the investigation as much 
as possible into distinct problems, the results of which may in many cases be taken for 
granted without the rest becoming unintelligible. In Part I. my object has been to put 
what maybe called the dynamical part of the subject in a clear light, deferring as much 
as possible special mathematical calculations. In the first place, I have considered the 
general theory of resonance for air-spaces confined nearly all round by rigid walls, and 
communicating with the external air by any number of passages which may be of the 
nature of necks or merely holes, under the limitation that both the length of the necks 
and the dimensions of the vessel are very small compared to the w 7 ave-length. To prevent 
misapprehension, I ought to say that the theory applies only to the fundamental note of 
the resonators, for the vibrations corresponding to the overtones are of an altogether 
different character. There are, however, cases of multiple resonance to which our theory 
is applicable. These occur when two or more vessels communicate with each other and 
with the external air by necks or otherwise ; and are easily treated by Lagrange’s general 
dynamical method, subject to a restriction as to the relative magnitudes of the wave- 
lengths and the dimensions of the system corresponding to that stated above for a single 
vessel. I am not aware whether this kind of resonance has been investigated before, either 
mathematically or experimentally. Lastly, I have sketched a solution of the problem of 
the open organ-pipe on the same general plan, which may be acceptable to those who are 
not acquainted with Helmholtz’s most valuable paper. The method here adopted, though 
it leads to results essentially the same as his, is I think more calculated to give an insight 
into the real nature of the question, and at the same time presents fewer mathematical 
difficulties. For a discussion of the solution, however, I must refer to Helmholtz. 
In Part II. the calculation of a certain quantity depending on the form of the necks 
of common resonators, and involved in the results of Part I., is entered upon. This 
quantity, denoted by c, is of the nature of a length, and is identical with what would be 
called in the theory of electricity the electric conductivity of the passage, supposed to be 
occupied by uniformly conducting matter. The question is accordingly similar to that 
of determining the electrical resistance of variously shaped conductors — an analogy of 
