264-265] GENERAL EQUATION. 481 



Also, writing p = p Q (1 + s) in the general equation of continuity, 

 Art. 8 (4), we have, with the same approximation, 



dt dx~ dy~ dz z &quot; 

 The elimination of s between (1) and (2) gives 



or, in our former notation, 



Since this equation is linear, it will be satisfied by the arith 

 metic mean of any number of separate solutions &amp;lt;/&amp;gt; ly &amp;lt; 2 , &amp;lt;/&amp;gt; 3 ,.... 

 As in Art. 39, let us imagine an infinite number of systems of 

 rectangular axes to be arranged uniformly about any point P as . 

 origin, and let &amp;lt;f&amp;gt; 1 , &amp;lt;f&amp;gt;, &amp;lt; 3 , ... be the velocity-potentials of motions 

 which are the same with respect to these systems as the original 

 motion &amp;lt;/&amp;gt; is with respect to the system x, y, z. In this case the 

 arithmetic mean (&amp;lt;/&amp;gt;, say), of the functions &amp;lt;f&amp;gt; l} &amp;lt;/&amp;gt; 2 , &amp;lt;/&amp;gt; 3 ,... will be 

 the velocity-potential of a motion symmetrical with respect to 

 the point P, and will therefore come under the investigation of 

 Art. 264, provided r denote the distance of any point from P. In 

 other words, if $ be a function of r and t, defined by the equation 



(5), 



where &amp;lt; is any solution of (4), and Biz- is the solid angle subtended 

 at P by an element of the surface of a sphere of radius r having 

 this point as centre, then 



dt* dr* 



Hence r$ = F(r-ct)+f(r + ct) .................. (7). 



The mean value of $ over a sphere having any point P of 

 the medium as centre is therefore subject to the same laws as the 



* This result was obtained, in a different manner, by Poisson, &quot; Me moire sur la 

 the orie du son,&quot; Journ. de VEcole Polytechn., t. vii. (1807), pp. 334338. The remark 

 that it leads at once to the complete solution of (4) is due to Liouville, Journ. de 

 Math., 1856, pp. 16. 



L. 31 



