796 Mr. S. Lees on the Analysis of 



Thus Xi 2 is the product o£ expressions (vii.) and (viii.), 

 whilst (vi.) becomes 



E * '=4j J f%i(p)%i(^) cos b(^+^/C) + o> 1 ] 



jr=0 p = 2=0 



4- cos [q{t+y/C)+a>i'] dy dp dq. (ix.) 



- £ f" f" f" *' (P) %ife) (cos [(p - 0)y/C] 



C7r Jo Jo Jo 



+ cos [(p + ^)(^+,y/C) + 2a) 1 ]}^^^. (x.) 



If we average this over a sufficiently large time, the time 

 term in this expression can be neglected. We may now 

 regard E x ' as the limit when y->co } of 



I 1 ! Xi(p) XM cos {p-9)y/Q • dy dp dq. (x. a) 



JoJo Jo 



167T ,„ |Q 



Integrating first with respect to y we obtain 



w> T+ « fT / , / xSin[(p-?)y/0] 7 , 

 E * »JS I^rJo Jo XAPj Xlig) (SF#7 ^ *' 

 We shall introduce a new variable ~= (p — q)y/C, so that 

 p=g.+ sCfr. 



We then have 



where the <? integration is left second. The expression 

 reduces to 





%)z=—qy, 



mere J/->^> . Since, however, 



sin z 



i 



we therefore get 



•) 



E/= § £" [ Xl (<?)] 2 . dg= g £" [ Xl (^)] 2 . 4»; (xii 



provided that %i(<?) is always finite, with a finite number o£ 

 discontinuities. A similar expression will obviously hold 

 for the energy due to the second integral in (iii.)- Hence 



