86 



c, (7) becomes 





V'7lOg p -V- ^ 



€ 



(0) 



(9) 



-/- 



I.O., In the present case of spherical syrunetry 



-^_ „ c ^^^;^ - « P (r,t) , say. 



Let us multiply the equation (9) by an arbltrainr function 

 S t -I- iill2j-I j and intecrate on t from -«> to and on r 

 from r^ to ©o, where r^< r, . (The function S will play the 

 role of the "DiraC delta function" used in quantum 

 mechanics. ) 



On integj^atinc by parts and taking <r and P as 

 ^rpnifV'lnr, at t s - oc or at r s oo , we obtain 



^ 'tsO /r!- -^ t«o 



P(rg-) cl dt 

 ?r 

 '-^ '" —T^v^ loo 



ili--c^4X 



9 t'- a r^ 



re- M-.\ dt 



F<fdtdr (10) 



But 



and 



in the third integral on the left we can use -|4- = c -^^ » 



dr 



so that (10) becomes 



d (re) _^ c ^(^<^) 



+ ^-^^ ^(l^)dr^cr„<r(r.,0)^(s^) 



,/cV^^^ 



r s- 



t.O 



5^' 



1T-| dt = y 



/ pi'dtdr 



(11) 



