1G9, 170.] GENERAL EQUATIONS OF SOUND WAVES. 207 



we have 



&amp;lt;r$_ d/ d$\ 

 r ~~ cr 





the solution of which is 



The mean value of &amp;lt;/&amp;gt; over a sphere having any point P of 

 the medium as centre is therefore propagated according to the 

 same laws as a symmetrical spherical disturbance of the air. We 

 see at once that the value of &amp;lt;/&amp;gt; at P at the time t depends on 



the mean initial values of &amp;lt;j&amp;gt; and ~ over a sphere of radius ct de- 



u/t 



scribed about P as centre, so that the disturbance is propagated 

 in all directions with uniform velocity c. If the disturbance 

 be confined originally to a finite portion S of space, the disturb 

 ance at any point P external to 2 will begin after a time 



-* , will last for a time - -, and will then cease altogether; 

 c c 



i\ t r a denoting the radii of the spheres described with P as centre, 

 the one just excluding, the other just including S. 



To express the solution of (23), already virtually obtained, in 



an analytical form, let the values of &amp;lt;/&amp;gt; and ~ , when t = 0, be 



eft 



* This result was obtained, in a different manner, by Poisson, J. de VEcole 

 Polytechniquf, 14 me cabier (1807), pp. 334338. The remark that it leads at once 

 to the complete solution of (23), first given by Poisson, Mem. de VAcad. des Sciences, 

 t. 3 (181819), is due to Liouville, J. de Math. 1856, pp. 16. The above 

 references are taken from Liouville s paper. 



The equation (24) may be proved also as follows. Suppose an infinite number 

 of systems of rectangular axes arranged uniformly about any point P of the fluid as 

 origin, and let &amp;lt;f&amp;gt; lt 2 , 3 , &e. be the velocity-potentials of motions which are the 

 same with respect to these systems as the original motion is with respect to the 

 system x, ?/, z. If then denote the mean value of the functions lf 2 , 3 , &c., 

 will be the velocity-potential of a motion symmetrical with respect to the point P, 

 and will therefore satisfy (12). The value of at a distance r from P will 

 evidently be the same thing as the mean value of over a sphere of radius r 

 described about P as centre. 



