482 WAVES OF EXPANSION. [CHAP. X 



velocity-potential of a symmetrical spherical disturbance. We 

 see at once that the value of <f> at P at the time t depends on 

 the mean initial values of <f> and d<p/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 2 of space, the 

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

 rj/c, will last for a time (r 2 r^/c, and will then cease altogether ; 

 r lt r z denoting the radii of two spheres described with P as centre, 

 the one just excluding, the other just including S. 



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

 an analytical form, let the values of (/> and d<f>/dt, when t = 0, be 



+ ~ + (x,y,z), - X (*,y,*) ............... (8). 



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

 r described about (a?,- y, z) as centre are 



<f> = ^ 1 1 ty (as + Ir, y + mr, z + nr) (for, 

 I T > y + mr,z + nr) dvr, 



= -^ 1 1 X 



where Z, ra, n denote the direction- cosines of any radius of this 

 sphere, and &cr the corresponding elementary solid angle. If we 

 put 



I = sin 6 cos ft), m = sin sin o>, n cos 0, 



we shall have 



&*= sin 



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

 </> at the point (x, y, z), at any subsequent time t, is 



1 rt r r 

 <f> = T -r. t 1 1 ty (# + ct sin 6 cos a), y -f ct sin sin ft), 



z -h ct cos 0) sin OdBdo) 

 + 1 1 x (x + ctf sin 6 cos ft), T/ + ct sin sin ft), 



2 + c cos 6) sin 0d0d&) . . . (9), 

 which is the form given by Poisson*. 



* " M^moire sur I'int6gration de quelques Equations lineaires aux differences 

 partielles, et particulierement de 1'equation generale du mouveraent des fluides 

 elastiques," Mtm. de VAcad. des Sciences, t. iii., 1818-19, 



