the Retarded Potentials and Huygliens s Principle. 1015 



suppose a sphere with centre at P to contract with the speed 

 of light so as to shrink to a point at P at time t. 

 Let us consider the integral 



HJ +-:$)*■' ■•••<*> 



where r = radius o£ the sphere, c = speed of light, co — solid 

 angle about P, and the surface of integration is the portion 

 Gr of the sphere that is momentarily included within S. 



Clearly at time t the last two terms become negligible (at 

 least, provided the first derivatives of cf> are all bounded or 

 limited in value) and I reduces to the value of <£ at P. 



-m 



We can obtain a connexion between I and p by considering 

 the change produced in I when the sphere contracts from one 

 position to another through a distance — dr = cdt. The 

 change in any term Z of the integrand is 



dt + ^-—dr or dt\^~ — c^— ). 



~dt 

 Hence 



the second integral extending only over the increment g 

 which is added to Gr by the displacement of the sphere. 



