188 The Rev. S. Earnshaw on the Triplicity of Sound. 



creases as k diminishes, while that of the other wave diminishes 

 under the same circumstances. An ordinary wave is therefore 

 essentially different from the other ; and the difference between 

 them may be made to rest on the properties here pointed out. 

 The second kind of wave is not and cannot be a wave of the 

 first kind. They may also be distinguished by this property — 

 that the length of a wave of the ordinary kind is always greater, 

 and that of a wave of the minute kind always less, than 2h, the 

 distance between two particles of the air. 



17. Equation (11) shows that for any value of v there is a 

 corresponding value of £; but as in the preceding investigations 

 k was taken to be essentially positive, the only admissible values 

 of v are those which lie between zero and V. And of these, 

 those which lie between V and ^V belong to the class of ordi- 

 nary gentle sounds, and those which lie between f V and to 

 the class of minute gentle sounds. 



18. Equation (8) teaches us that when the sound is of the 

 type which we have termed violent, any value of v is possible 

 between V and Qo. And hence the conclusion with respect to 

 possible velocity is, that the atmosphere is capable of transmitting 

 sound-waves with any degree of velocity from zero to infinity. But 

 it must be noticed that this rauge of velocity divides itself into 

 three essentially distinct portions ; viz. from to |V, from |Vto V, 

 and from V to x, corresponding to the three distinct kinds of 

 waves — minute , ordinary, and violent : and the two former belong 

 to the circular, and the last to the exponential type. And with 

 the same value of k there may coexist three distinct waves — one of 

 each of these kinds — all propagated with different velocities. This 

 is what is meant by the title at the head of this communication, 

 the triplicity of sound. 



19. The form of equation (11) shows that k admits of a maxi- 

 mum by the variation of v, viz. ^. (It is easily seen that it is 



the velocity corresponding to this value of k which separates the 

 two classes of gentle sounds.) Hence we perceive that the par- 



V 



tides of the atmosphere cannot execute more than -rj free vibra- 



tions per second. 



20. Much additional light will be thrown on the results 

 arrived at in the preceding articles with respect to the wave-pro- 

 perties of an elastic medium like the atmosphere, if we present 

 the relation between k and v to the eye by means of curves. 

 There are two types of waves — the circular and the exponential. 

 In the case of the former the analytical relation between k and v 

 is expressed by the equation (11). We may show this by a 

 curve, if we take the different values of kh for the abscissae, and 



