215 



APPEfDIX A. 



The proof of the relation jiven in equation (3) follows analogous lines to the proof of 

 Kirchoff's general solution of the wave equation jiven by Jeans, "The hatheinatical Theory of 

 Electricity ana Magn?tism", nth Edition, paragraph 590, pajes 622-52U. Equation (5U3) of Jeans, 

 with slijhtly changea notation, is 



\ 



- t 

 t 



p is - G Lp; q S 

 9 n 3 n 



J J 



p Ls - G Le 



3 t 3 t 



ax ay jz 



(Al) 



- f 



Where p and G are any two solutions of the wave-equation, the summation is taken over all the 

 surfaces Dounding the region of the volume integral ana in which the wave-equation holds, n is the 

 normal to the surface drawn into this region and t" and - t' are positive and negative times 

 respectively. 



Since G can be any solution of the wave-equation we take it to be 



F' (ct * r ) _ F (ct * r) 



r 



ct * r) I 



Cos e 



(A2) 



L5 



d n 



B n 



5_ 



r 3 n 



_2p 3_r ^ 1 3_p 

 r 3 n r 3 n 



F- (ct + r) Cos 



P |i F- (ct + r) Sine 

 r 3 n 



2p 3 r . 1 3 



I£ 11 * J 

 r^ 3 n r 



2_E 

 3 b 



F (ct + r) Cos I 



* 4 1* F (ct + r) Sin 5 

 r 3 n 



(A3) 



Integrating the first three terms of (A3) by parts, we obtain, 

 t" 



i ILC Cos<9 



- f 



p F" (ct + r) 3t = — ^- Cos 

 cr 3 n 



p F' (ct + r) - -4^2- F (ct * r) 

 c 3 t • 



3 r 

 r 3 n 



Cos 



3^0 



2 ^P r F (ct + r) dt 

 c 3 t*^ 



(A4) 



Cos 



f 



ifiLl * L£ |. F' (ct * r) at = -££S.^ 

 r 3 n 3 n 



2£ Lr ♦ |j F (ct + r) 

 r 3 n 3 n 



+ Cosg 

 cr 



rdnBt 3n3t 



(a5) 



Sin ^ 



