SIMPLE-HARMONIC WAVES. DIFFRACTION 235 



If we eliminate u, v, w by the kinematical relation (1) of 70 

 we obtain 



(8) 



If X, Y, Z, s all vary as e ikct , this becomes 



V*s + k*s = \div(X, Y,Z), ............ (9) 



c 



in the notation of 67. This is of the form (3), and the 

 solution is therefore 



1 niftX' ar dZ'\e- ik ' ,, ,,, 



8 = - A ; -*-T + -5-7 + *-i) - dxdy dz' 

 4>7rc z JJj\dx dy dz J r 



The transformation is effected by partial integration of the 

 several terms, the integrated portions vanishing at infinity if 

 X', Y 7 , Z' do so. Also, since 



r = J[(x - xj +(y- yj + (z- 



we have 

 whence 



- <> 



From this the value of ^> follows by the relation 



(12) 



As a particular case, suppose that Y' = 0, Z' 0, and that 

 X' differs from only in a small region about the origin, 

 and put 



pjjjX'dx'dy'dz' = P ................ (13) 



We have < = - r ^- j- (} , ............... (14) 



dx\ r J 



or, for large values of kr, 



P p-ikr 



cos(9, ............... (15) 



r 



as before. Comparing with 76 (24) we see that a concentrated 

 force Pe int has the effect of a double source of strength iP/pkc. 



