Small Vibratory Motions of Elastic Fluids. 283 



according as the propagation is from or towards the origin of .r. 

 If in consequence of a limited duration of the disturbance, the 

 number of oscillations of each particle be limited to n, the 

 preceding equations- are apiilicable at a given instant, only to 

 particles included in a space n\, all the others being at rest, 



and to a given i)article only for an interval — , which is the 



duration of its motion. In general, having given the commence- 

 ment and duration of a disturbance, supposed to cause a certain 

 number of oscillations exactly, we can always determine what 

 particles are affected by it at a given instant ; also the beginning 

 and end of the motion which a given paiticle undergoes. 



Suppose now that there are several disturbances of the same 

 kind as that above considered, and that some produce motion 

 in the positive direction, some in the contrary. Having given 

 the commencement and duration of each disturbance, it is re- 

 quired to find in what manner the particles are moving at any 



given instant. The differential equation -y^ = a" -— |- will be 

 satisfied if we put 



(j> ^ F, {x - at) + F, (.r - at) + &c. 

 + /, {.V + at) + /„ (.r + «0 + &c. 



each of the functions belonging to a separate disturbance. If 

 we consider each function by itself, it will give us the motion 

 which results from the disturbance to which it belongs. Hence 

 the above equation shews that when a particle is affected by 

 several disturbances simultaneously, the motion it receives is the 

 resultant of all the different motions it would have, if each dis- 

 turbance acted separately. And this is a general proof of the 

 co-existence of small vibrations, in rectilinear propagated motion. 

 The problem in question will therefore be solved, by putting 



NN 2 



