Impulsive Motion of Electrified Systems. 145 



If w be infinitesimal compared with v—u, (49) becomes 



n\ — \i) w eos7~| 



Kj 



w 



■-h- 



(50) 



Applying (50) to a sphere of radius a with a uniform sur- 

 face-charge Q in the manner explained in § 7, we find that 

 in the pulse, a<t a great distance from the sphere, 



B= QF-' + ^-^r y i. . . (si) 



2Kra L v/i r-/ r J v 



]• (32) 



If yfr be tlie angle between u and w. we find for the energy 

 per unit volume in the pul>e, since fiKv 2 =l and a=u/v, 



KE 2 _ fiQ 2 w 2 [~1 2n cosy cos -xlr (1— ?i 2 )cos 2 7 

 ~4^~~l(J7nV 2 U? + ¥ ! " ~A~ 



One angular coordinate of r is 6 ; if the other be $ and 

 be measured round the direction of u from the plane of u and 

 w } we have 



eos7 = cos #cos -v/r-fsin 6 sin -^ cos <£. . (53) 



The various angles are shown in fig. 5. 



The thickness of the pulse is 2a, and 

 hence the energy radiated away in the 

 pulse is given by 



W = jT J ~ 2a r- sin 6 dO d(f>. 



Fig-. 5. 



From (53) we find 



r 



cos 7 defy 



27TCOSo/r(l — h) 



r 



cos 2 7 # = ^ { (2-3 sim»(l - h) 2 + ft 2 sin 2 i/r}, 



Integrating with respect to </> from to 2tt, and writing 

 dh/n for sin (10, we find 



W = A-Bsin 2 ^. 



* ttOVf/ 1 2 l-w 2 \ 7/ 



A = wjl-ip+jp-np-)*' 



l— » 



/iQ-W T r _ 3 + ^ 2 3-ri 

 4<m 3 J 1 2 A 2 ~*~ A 3 



3-» 2 (l-rc 2 )(3-n s 



2A 4 



Irf/*. 



Phil. Mao. 8. 6. Vol. 13. No. 73. /an. 1907. 



