﻿240 Royal Society. 



class of resonators, including most of those which occur in practice. 

 When a mass of air or other gas is enclosed in a space bounded 

 nearly all round by rigid walls, but communicating with the external 

 air by one or more passages, there are certain natural periods of 

 vibration or resonant notes whose determination is a matter of in- 

 terest. If the dimension of the air-space is small compared with the 

 wave-length of the vibration, the dynamics of the motion is, in its 

 general character, of remarkable simplicity. It is for the most part 

 under this limitation that the problem is considered in the present 

 paper. The formula determining the resonant note is 



iV 



where n is the number of complete vibrations per second, a the ve- 

 locity of sound, and S the capacity of the air-space ; c is a quantity 

 proved to be identical with the measure of electric conductivity be- 

 tween the interior of the vessel and the external space, on the sup- 

 position that the air is replaced by a uniform conducting mass of 

 unit specific conducting-power, and the sides of the vessel and 

 passages by insulators. When there is more than one passage, the 

 formula is still applicable according to the above definition of c ; and 

 when the passages are sufficiently far apart not to interfere with each 

 other, the resultant c is, by the electrical law of parallel circuits, 

 simply the sum of the separate values for each passage considered 

 by itself. When this condition is not satisfied, the value of c, thus, 

 found by mere addition, is too great. 



The question thus resolves itself into the determination of the 

 conductivity (or the resistance which is its reciprocal) for different 

 forms of passages or openings. The case of openings, which are 

 mere holes in the sides of the vessel, has been .already treated, 

 although in a very different way, by Helmholtz, who, in his cele- 

 brated paper on vibrations in open pipes, compared his theory with 

 the observations of Sondhauss and others on the notes produced 

 when such resonators are made to speak by a stream of air blown 

 across the mouth. Sondhauss has also given an empirical formula 

 applicable when the connecting passages are of the form of long cy- 

 lindrical necks. These previous results are in agreement, as far as 

 they go, with the formula here investigated, and which is applicable 

 whatever may be the length of the neck. If L be the length and R the 



1 L + 2 R 



radius - or tne electrical resistance = — — — . 



C 7T li 



This supposes the neck a circular cylinder. If the section be an 

 approximate circle of area <r, we may put 



r-7 + 2\/; 



When the neck is very long, the second term may be neglected ; 

 and when L is very small, the first term becomes insignificant. In 

 the third part experiments are described which were instituted to 



