Y. On the Theory of Resonance. By the Hon. J. W. Strutt, 31. A., Fellow of Trinity 
College , Cambridge. Communicated ly W. Spottiswoode, F.R.S. 
Received July 2, — Read November 24, 1870*. 
Introduction. 
Although the theory of aerial vibrations has been treated by more than one generation 
of mathematicians and experimenters, comparatively little has been done towards obtain- 
ing a clear view of what goes on in any but the more simple cases. The extreme diffi- 
culty of any thing like a general deductive investigation of the question is no doubt one 
reason. On the other hand, experimenters on this, as on other subjects, have too often 
observed and measured blindly without taking sufficient care to simplify the conditions 
of their experiments, so as to attack as few difficulties as possible at a time. The result 
has been vast accumulations of isolated facts and measurements which lie as a sort of 
dead weight on the scientific stomach, and which must remain undigested until theory 
supplies a more powerful solvent than any now at our command. The motion of the 
air in cylindical organ-pipes was successfully investigated by Bernoulli and Euler, at 
least in its main features ; but their treatment of the question of the open pipe was 
incomplete, or even erroneous, on account of the assumption that at the open end the 
air remains of invariable density during the vibration. Although attacked by many 
others, this difficulty was not finally overcome until Helmholtz f, in a paper which I 
shall have repeated occasion to refer to, gave a solution of the problem under certain 
restrictions, free from any arbitrary assumptions as to what takes place at the open end. 
Poisson and Stores J have solved the problem of the vibrations communicated to an 
infinite mass of air from the surface of a sphere or circular cylinder. The solution for 
the sphere is very instructive, because the vibrations outside any imaginary sphere 
enclosing vibrating bodies of any kind may be supposed to take their rise in the surface 
of the sphere itself. 
More important in its relation to the subject of the present paper is an investigation 
by Helmholtz of the air-vibrations in cavernous spaces (Hohlraiime), whose three dimen- 
sions are very small compared to the wave-length, and which communicate with the 
external atmosphere by small holes in their surfaces. If the opening be circular of area 
tf, and if S denote the volume, n the number of vibrations per second in the fundamental 
* Additions made since the paper -was first sent to the Royal Society are inclosed in square brackets [ ]. 
t Theoric der Luftschwingunsren in Rohren mit oflfenen Enden. Crelle, I860. 
7 Phil. Trans. 1868, or Phil. Mag. Dec. 1868. 
MDCCCLXXI. 
M 
