4o Mr. G. F. C. Searle on the Impulsive 



Q 2 /2«K = U , the electric energy of the sphere at rest, we 

 have 



w jl-n^cosaf «fe 2 }.(6) 



l - Z7T J (1 — ?i 1 cos^ 1 )(l— n 2 cosv 2 ) J 



The only difficulty lies in the evaluation of the integral in 

 this expression. Since the integral is a function of n iy n 2 , 

 and a, we shall write 



F(w 1? U 2 , a)= \ r , n^y-. -x-r, . (7) 



y J (l — ^i cos 6 l ){l — n 2 cos 6 2 ) 



where the integration extends over the surface of the unit 

 sphere of which dco is an element. Thus 



W=U { (l-*,«a cos «)(2 7 r)- 1 F(n lj n 2 , «J-2}. (8) 



§ 4. When the directions of Uj and u 2 are parallel and in 

 the same sense, so that cosa=l, the value of F is easily 

 found, for we now have 9 2 = 6 X and can take 2tt sin l cW 1 as 

 the element of surface of the unit sphere. Thus 



BV n\ 9 i" sinW 



■E>i%,0> = ^J o (i_, ilC os^)(l-» 2 cos^) 



2t J C" sin (9,^ f * sinAi^x 



ftj — n 2 I 1 J g 1 — <i 1 cosc' 1 2 J 1— n 2 cos0 1 



= ^L_ log a+"oa-'") (9 ) 



When u x and u 2 are parallel but directed in opposite direc- 

 tions, so that cos a.— — 1, we have cos 2 = — cos 19 and hence 

 the value of F can be found from (9) by changing the sign 

 of n 2 . Then 



IfWj, ?2 9 , 7T) = loo-)- if)- ^. . (10) 



' ^H-Wg & (1 — ?i 1 )(L— ?? 2 ) 



Substituting these values of F in (8) we obtain the following- 

 expressions for the radiated energy : — 



When u x and u 2 are parallel and in the same direction, so 

 that cosa=l, 



W=U {*=** log g +w ffir"l -2 } . • (11) 



