130 Mr. G-. F. C. Searle on the 



Hence the impulse required to start the system exceeds that 

 required to stop it by twice the momentum carried away by 

 the pulse in either case. 



§ 17. We will now apply the principles explained in the 

 earlier part o£ this paper to some simple systems, and will 

 calculate the energy and momentum in the pulse formed when 

 the system is set into motion at speed u. By § 7, the electric 

 and magnetic energies in the pulse are equal, and thus the 

 total energy in the pulse is ultimately equal to 



where dco is an element of solid angle. Hence, by (11) ,. 

 since u/v = n, we have 



• ' w =gjJ^SKS) 2 - • • • (34 > 



In the pulse, E and H are perpendicular to the radius r 

 and to each other. Hence, attending to the directions of E 

 and H, we see that YEH is in the direction o£ the outward 

 radius and that its magnitude is simply EH. Hence, by (16), 

 if v l denote a unit vector in the direction o£ the radius, the 

 momentum in the pulse is given, in vector notation, by 



(35} 



_ llK CC ^ rT o 7 7 l xun i i r i sm 2 dco dz fdQy 



p = £L- 1 1 r E II r 2 dco dz =. V- 1 \ 7 \ n ^ -^ ) . 



4tt JJ j 4tt J J (1 — n cos 0Y \dz J 



Here and in (34) the range of z is from p l to _p 3 , where 

 P2~~Pi—P> ^ ne thickness of the pulse. 



The component of r x in the direction of u is cos 6, and 

 hence when, for any reason, P is parallel to u, it has the value 



p_ /Mm rr sin 2 6 cos 6 dco dz I dQY 2 , q ,. 



~ ^ J J (1-ncosVf \dz~J ' ' * W 



When the system is symmetrical round an axis parallel to 

 the direction of motion, we may put dco — 2tt sin dd, and 

 then 



w_A«rPP sin 3 6 d6 dz /dQX 2 



T JJ (i-ncosey{dz) ' 



n\0 cos cW d? /di 

 (l-ncos(9)V \dz) 



pun CC sin 3 6 cos cW dz fdQ^ 2 



(37) 

 (38) 



where goes from to ir. 



§ 18. 'Small Velocities. — Let the radius be defined by 9 



