Small Vibratory 31otions of Elastic Fluids. 311 



tances. Hence supposing the number of the pyramids indefinitely 

 great, and the velocities and condensations at equal distances to 

 be given at a given instant by any law, depending on the initial 

 disturbance, it will be possible to calculate the circumstances of 

 the motion at any instant. Again, though the general character 

 of propagation in space of three dimensions is spherical, it is 

 not necessary that the boundary of a wave should be a spherical 

 surface. It may be any surface whatever: but we infer from 

 the general projierty, that each very siriall portion of the wave 

 will obey the laws of spherical propagation: — it will move uni- 

 formly in the direction of its normal, as if it were a portion of 

 a spherical wave having for its centre the centre of curvature. 

 It is easy to see that the centres of curvature will be fixed 

 points in space, and that the surface, whatever it may be at first, 

 will continually approach to that of a sphere. The reasoning in 

 this Art. is exactly analogous to that which is stated at the end 

 of Art. 10. 



The principle of the conservation of vis viva will hold in 

 every wave, whatever be the shape of its boundary or the law 

 of its type, because it holdfe for every individual portion of it. 



17. Let us now make an application of the general solution 

 to an instance in which the disturbance is of a very general 

 nature. Suppose given disturbances to act at any number of 

 points in space for any length of time: it is required to find 

 what will be the consequent motion at a given instant of a par- 

 ticle in a given position. We will assume fixed rectangular axes 

 and an origin of x, y, z; and date t from a fixed epoch, l^ei 



be the co-ordinates of the points of disturbance; t,, t„, t^, &;c. 

 the intervals between the commencements of the disturbances 

 and the beginning of t. Then the solution of the question will 

 be effected by the equation, 



