578, 579] Equation of Wave-propagation 503 



For small values of r this assumes the form 



X = \ [{/(at) + fc (a*)} - r {/' (at) - V (at)} + {/" (at) + 3>" (erf)} + . . .] 



......... (539). 



In order that X may be finite at the origin through all time, we must 

 have 



f(at) + < (at) = 



at every instant, so that the function <I> must be identical with /. On 

 putting r = 0, equation (539) becomes 



and from equation (537), putting r = 0, we have 



so that 4nr(tiV- =-2/ / (a) ........................ (540). 



Equation (538) may now be written as 



rX =f(at - r) -f(at + r). 

 On differentiating this equation with respect to r and t respectively, 



j-(r\)=:-f'(at-r)-f'(at+r), 



HtW- f'(at-r)-f(at + r), 

 and on addition we have 



. 

 This equation is true for all values of r and t : putting t = 0, we have 



as an equation which is true for all values of r. Giving to r the special 

 value r = at, the equation becomes 



The left hand is, by equation (520), equal to 4<7r(u) r=0 . If we use u, u to 

 denote the mean values of u and u averaged over a sphere of radius at at 

 any instant, the equation becomes 



(u) r=0 = ^ (tu t=0 ) + ^ =0 ..................... (541). 



Thus the value of u at any point (which we select to be the origin) at 

 any instant t depends only on the values of u and u at time t = over a 

 sphere of radius at surrounding this point. The solution is of the same 

 nature as that obtained in 578, but is no longer limited to spherical waves. 



