54 Mr. G. F. C. Searle on the Impulsive 



Using (29), we easily find that when #=B = (1— %)"* 



Xi = n 1 (l-n 2 \)(l-n 1 )-\ 

 and that when # = A= (l-f^) -1 



X5=w 1 (l + w 2 \)(l + Wi)- 1 . 

 Thus 



Y _ (1 +^x) { p 2 —n 2 2 + ftiflgA, 4- n 1 n 2 2 —n 1 2 n 2 \ +pn 1 {l—n 2 \) } 



(1 — n 1 )^p 2 ~-7? 2 2 + n x n 2 \— n^f + ni 2 w 2 X + pii(l -f n 2 A.) £ 



When we substitute for p 2 from (29), we find that both 

 numerator and denominator contain n x as a factor. When 

 this factor is removed, we have 



Y _ (1 + ni) {ni + n 2 2 —n x n 2 2 — ?z 2 A — 7z 1 w 2 \ + n 1 n 2 2 X + y>(l — ?2 2 \) } 



(1 — 7?j) {n!— n 2 2 -~n 1 n 2 2 — n 2 X -+- w^A, + n 1 n 2 2 X +p0- — n 2 X) } 



On carrying out the multiplications, we find that the five 

 terms on the right side of (29) occur in both numerator and 

 denominator. Replacing these terms by p 2 , we obtain 



v _ (l—n 1 n 2 \)(n 1 — n 2 \)+p(l—n 1 n 2 \ + 7ii--n 2 7C)+p 2 

 (1 — - W]?i 2 A,) '(ni — n 2 \) -f p (1 — riin 2 \ — n x + w 2 \) — p 2 

 __ ( l — n 1 n 2 \ + p )(n 1 — y 2 \+p) 



~ (1 — Jl^A, — JO) (W! — W 2 A, -f p) 



_ 1— n 1 ft 2 \+£> 

 ~~ 1 — riin 2 \ — p' 



Since, by (29) and (27), p 2 = m 2 — ?? l 2 ^ 2 2 sin 2 a and since 

 X = cos a, we obtain, finally, 



F(w x , ra 2 , «) = 



27T , 1 — n!?? 2 cos a -\- (m 2 — n^n 2 2 sin 2 a)* 



(m 2 —n^n 2 2 sin 2 a)* ° 1 — %?z 2 cos a— {m 2 — n^n 2 2 sin 2 a)* * 



This result agrees with that found in § 9. 



§ 11. We can now return to the electrical problem and 

 obtain an expression for the energy radiated when the velocity 

 of the sphere is impulsively changed. 



Since, by (8) ' 



W=U {(27r)- 1 (l-w 1 ?2 2 cos a )F(?2 1 , » 2 , a)-2} 

 we have, by (28) 



w/u = 



1 — %ra 2 cosa , 1— n^ cos x. + (m 2 — u^u^ sin 2 a)* 

 (m 2 — % 2 rc 2 2 sin 2 a)* ° 1 — w^ cos a —• (m 2 — n^n 2 2 sm* a)* 



