482 WAVES OF EXPANSION. [CHAP. X 



velocity-potential of a symmetrical spherical disturbance. We 

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

 the mean initial values of (/&amp;gt; and d(f)/dt over a sphere of radius ct 

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

 in all directions with uniform velocity c. Thus if the original 

 disturbance extend only through a finite portion S of space, the 

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

 r-i/c, will last for a time (r 2 ?*i)/c, and will then cease altogether ; 

 r-L , r 2 denoting the radii of two spheres described with P as centre, 

 the one just excluding, the other just including 2. 



To express the solution of (4), already virtually obtained, in 

 an analytical form, let the values of (f&amp;gt; and d(f&amp;gt;/dt, when t = 0, be 



&amp;lt;j&amp;gt; = ty(x,y,z\ -* = x (a;,y,*) ............... (8). 



The mean initial values of these quantities over a sphere of radius 

 r described about (#, y, z) as centre are 



&amp;lt;/&amp;gt; = -^ 1 1 ty (x + Ir, y + mr, z + nr) dvr, 

 = 



x + r &amp;gt; y + mr &amp;gt; z + nr &amp;gt; 



where I, m, n denote the direction- cosines of any radius of this 

 sphere, and 57 the corresponding elementary solid angle. If we 

 put 



I = sin 6 cos ft&amp;gt;, m = sin 6 sin o&amp;gt;, n cos 6, 



we shall have 



Stzr =sin 



Hence, comparing with Art. 264 (15), we see that the value of 

 (f&amp;gt; at the point {x, y, z), at any subsequent time t, is 



&amp;lt;f&amp;gt; T~ -J-. t 1 1 ty ( + ct sin 6 cos &&amp;gt;, y -f ct sin 6 sin &&amp;gt;, 



z -\- ct cos 6) sin OdOda) 

 + 1 1 ^ (x + ct sin 6 cos o&amp;gt;, y + ct sin # sin o&amp;gt;, 



s + cZ cos 6) sin 0d0d&&amp;gt; . . . (9), 

 which is the form given by Poisson*. 



* &quot; M6moire sur I int6gration de quelques Equations lineaires aux differences 

 partielles, et particulierement de 1 equation generale du mouvement des fluides 

 elastiques,&quot; N.6m. de VAcad. des Sciences, t. iii., 1818-19. 



