Motion of an Electrified Sphere. 65 



This result is identical with (43), which was obtained in 

 § 14 by dynamical principles. 



§ 16. When w, the change of velocity, is small compared 

 with (v 2 — u 2 )i, where 2u is the resultant o£ u x and u 2 , we can 

 obtain approximate expressions for W and for If which may 

 be useful for many purposes. 



The energy radiated when w is small is easily deduced 

 from the expression given in § 12. Since, when yjx is less 

 than 1, 



* x — y \5x- ox / 



x 



y -x-y 



it follows that, when m or w/v is small, 



,2 a ; n v 



w _ 2m%l-n 2 sm*ylr) 

 ~ ° 3(l-n 2 +£m 2 ) 2 ' 



But U =Q 2 /2Ka = juu 2 Q 2 /2a, and thus as far as terms 



m iv~ 



w = 2lW(l-tt 2 sin 2 ^) /xQV 2 (l-?? 2 sin 2 ^) 

 3v 2 (l-w 2 ) 2 3«(l-?i 2 ); 



a value identical with that obtained when w is treated as small 

 throughout *. Thus the effect of a given change of velocity 

 depends upon the initial velocity. If W be the energy 

 radiated when the velocity is changed from to w, we havef, 

 as far as tu 2 , 



da 



and hence 



W _ 1— n 2 sin 2 i|r 



w,~ (i-n>)» • 



The following table shows the value of W/ YV f or a few 

 values of n and of yfr. 



It will be seen from the table that the angle i between the 

 initial velocity and the change of velocity has only a small 

 effect when the initial velocity is small. For initial velocities 

 less than ^v the energy radiated does not differ by as much 

 as 10 per cent, from the energy radiated when the initial 



* Phil. Mag. Jan. 1907, p. 146. 



t Phil. Mag. Jan. 1907, p. 131 (or § 13 above). 



Phil. Mag. S. 6. Vol. 17. No. 97. Jan. 1909. F 



