1G7, 168.] SPHERICAL WAVES. 205 



subsequent time t is then to be found as follows. Comparing (14) 

 with (13) we find 



rr 

 Integrating the latter equation, and putting I r%(r)dr = Xi( r }&amp;gt; we 



obtain 



- l , 



\j 



where C is an arbitrary constant This gives, in conjunction with 

 (15), 



, 



(16), 



The complete value (13) of &amp;lt;f&amp;gt; is then found by writing, for r&amp;gt; 

 r ct in (16), and r + ct in (17), viz. 



0+ixt(r--M ......... (I 8 )- 



I/ 



It is obvious from the symmetry of the motion with respect to 

 the origin that 



*(-r)= *(r), %(-r)= X (r) ............ (19), 



and therefore 



We shall require shortly the value of &amp;lt; at the origin. This 

 may be found by dividing both sides of (18) by r, and evaluating 

 the indeterminate form which the right-hand member assumes for 

 ? =0; or more simply by differentiating both sides of (18) with 

 respect to T and then making r = 0. The result is, if we take 

 account of the relations (19) and (20), 



(21). 



