Motion of an Electrified Sphere. 45 



If Q Y and 2 be the angles between the radius and the direc- 

 tions of u x and u 2 , we have u 1 R = ?j 1 cos 6 X and u 2 R=w 2 cos 2 , 

 and u 1 u 2 = if 1 ?i 2 cos a, where a is the angle between Uy and u 2 . 

 Thus we find 



-p, Q 2 J ui 2 sin 2 0! i< 2 2 sin 2 



4KW I (v — it! cos 0i) 2 (v — u-2 cos # 2 ) 2 



WlW 2 COS a — WjW 2 COS 0t cos 6 





(v — Hi COS #i)(t' — W 2 COS 2 ) 



The numerator of the last term in this expression can be 

 "written in the form 



2{v(v—v l cos l ) + v(y — u 2 co$0 2 ) — (fl— «i cos0 1 )(i < — ?/ 2 cos# 2 ) 



— u 2 +w 1 M 2 cosa}, 

 and thus we obtain, after an easy reduction, 



E 2 = Q 2 { 2(r 2 -z/ 1 i/ 2 cos«) 



4K-V 2 (( 2 I (/"— Mj cos0,)(v — 2f 2 C0S 2 



ir — Wi 



) (l'~ WiCOS^j) 



2 2 



ir — ?/ 2 



(v—i^cos o 2 y 



(3) 



Iii the pulse, the magnetic energy per unit volume is equal 

 to the electric energy per unit volume, and hence, remem- 

 bering that the thickness of the pulse is 2a, we find that W, 

 the radiated energy, is given by 



W-^flW., (4) 



where dco is an element of solid angle. 



The last two terms in the expression for E 2 can be inte- 

 grated at once. Thus we have 



J (v — u x cos V J 2 _ "jo {i — «i cos &i) 2 



V 2 — 2({ 2 ' 



The expression for TT now becomes 



w _ Q 2 | f 2(v 2 — u-jU 2 cos a)d<0 1 



87raK I J (r — ?/ x cos Oi)(y— i/ 2 cos # 2 ) J" 



Writing i^ for w a /t> and rc 2 for u 2 /v, and noting that 



