70 On the Impulsive Motion of an Electrified Sphere. 



The radiated momentum is, by symmetry, along the di- 

 rection bisecting the angle between the initial and final 

 velocities. In this case the two components P x and P 2 , 

 which are parallel to u x and u 2 , have equal values, and 

 thus 



P = 2PiCOS \a. 



Hence, since M^Mj and W 2 =Wi, we find by § U 

 that j 



V 2f0 ., M.Ccos^-lHWM 



Jf = L COS o« — ; 



1 + COSa 



Hence 



= (^--2M 1 sin 2 |a\secia. 



^r^-* sin4tt i sec|a , . . (46 > 



Uo L«iLo Lo J 



If we wish, we can express P in terms of mov by means of 

 the formula Uo/i' = 3m r/4, w r here m is the electromagnetic 

 mass for infinitesimal speeds. 



If we make use of the series (39) for M x and combine it 

 with the series (45) for W/U , we obtain the series 



Pi- 

 ll 



o l^o O . . i 



+ (85-104S 2 + 40S 4 ) 3^ -- +. . . A, 



■where, as before, S = sin^a. This series is certainly valid 

 when 7i! < 0-414 



In Table VIII., which was prepared by Mr. A. J. Bamford 

 of Emmanuel College, the values of Pv/Uo for %=0'1, ?i = # 2,, 

 and 7i 1 = 0'3 were calculated by aid of this series. For the 

 remaining values of n l5 Pr/U was calculated by (46) from 

 the values of "W/Uo given in Table VII. and those of Mit'/Uo- 

 given in Table II. 



When « = 7r, so that the motion is just reversed, the 

 value of P given by (46) becomes indeterminate since 

 then cos^a = and W=2i*iM l3 as appears from (13) and 

 (39). But in this case symmetry demands that P should 

 vanish. 



