690 BBPOBT — 1893. 



tliat tlie frequency of vibration (N) corresponding to n nodes in the length L 

 will be 



N 



-L^')- 



For instance, in a trough of deep water, when V oc '^\ 



N oc ^Jn. 



An example of a system with a remarkable reiation between the velocity of pro- 

 pagation and the wave-length is the case of a system of magnets with their poles 

 close to one another when disturbed to an amount small compared with the 

 distance apart of the poles. In this case the force of restitution is proportional to 

 the sum of the angular displacements of contiguous magnets, and it appears that 

 the velocity of propagation of a disturbance is given by 



V = 2N„\cos^^, 



A 



when X is the wave-length of the disturbance. This case is interesting in connec- 

 tion with Professor Ewing's theory of the nature of magnets, and it follows that 

 the rate of progression that might be expected of this kind of disturbance in 

 a real magnet would be extremely slow unless the period of vibration approxi- 

 mated to lO'' per second. The rate of propagation of energy into the system is, 

 however, very much more rapid, and might be about 200 to 300 centimetres per 

 eecond. In a finite system of such magnets the system of overtones is given by 



N = 2NoCos^^, 



Ij 



which is evidently represented by a series of lines coming up to an edge which is 

 so characteristic of many spectra. 



Vibrating linear systems having any desired relation out of a very great 

 number of different relations connecting the velocity of propagation of a wave and 

 the wave-length may be constructed by connecting a system of equidistant wheels 

 by means of indiarubber bands or elastic friction wheels, the latter case being some- 

 what similar to the case of the magnets already considered. By connecting the 

 wheels each with its next neighbour we get the simplest system. If to this be 

 superposed a system of connection of each with its next neighbour but one, and 

 then each with its next neighbour but two, and so on, complex systems with very 

 various relations between wave-length and velocity can be constructed depending 

 on the relative strengths of the bands employed. These systems would be some- 

 what analogous to systems of particles connected by laws of force varying in a 

 complex way with the distance apart of the particles. In the case of the bands, 

 &c., the general form of the relation between the velocity and wave-length is 



' X ° X ' X 



which can be varied in a very great number of ways by a proper choice of /;,, A;,, &c. 

 It was pointed out how a model such as that described by Mr. Glazebrook for 

 illustrating anomalous dispersion could be modified so as to produce almost any 

 desired system of overtones. 



5. On tlie Eefledion of Sound or Light from a Corrugated Surface. 



By Lord Ratleigh. 



The angle of incidence is supposed to be zero, and the amplitude of the inci- 

 dent wave to be unity. If then 



^=ccoapx (1) 



