208 



z. Then spheres with a radius r = ct are continually cutting the 

 domain of the original disturbance of equilibrium. Working out 

 the calculation we find that the number of integrations to be executed 

 if one of the coordinates does not occur is still the same as when 

 it occurs. ') That is the reason why in R^ a disturbance 

 of equilibrium never vanishes there where it once appeared. In 

 an analogous way we can pass from a solution for /?2;i+i to one 

 for R^n- III this way it becomes clear that the continuation of a 

 disturbance of equilibrium is a common property of all /?2»'s. 



IV. The easiest way to find these solutions is by means of 

 Kirchhoff's method.-) A special solution x of the equation without 

 right-hand side is then used. This •/ is a function of / and of the dis- 

 tance r to a fixed centre P only so that the equation which is satisfied 

 by X, becomes in Rn 



1 d'x n--l ÖX ö^x_ 



r bt^ r dr dr^ 



1 d 

 Applying the operation D = -^ to a special solution of this 



?■ Or 



equation we find a solution of the same equation for 7i-\- 2 instead 



of for n. For odd values of n the special solution is 



n—l . 



y^ = ]J ^ \G{t + - 



viz. for n = 1 : 



'(■n) 



where G is an arbitrary function; 

 for n ^ 3 : 



\ dG 1 , 



- -r~ or also :=! F [ t 

 r or r \ c 



{F an arbitrary function) ; 

 for n = 5: V 



r Or \r \ cJJ r \ cj r'c \ c 



etc. 



Applying Green's identity to the required solution ip and this x (e.g. 

 for n = 5) in the whole space tu outside a small sphere with 

 radius R round P we find 



1) Gomp. e.g. H. A. Lorentz: The theory of electrons. Note 4, p. 233. 



2) See e.g. Rayleigh, Theory of Sound, Ch. XIV, § 275. 



