Motion of an Electrified Sphere. 55 



In terms of the velocities the expression becomes 



v 2 — u Y u 2 cos a , v 2 — u 1 u 2 cos a + (y 2 w 2 — u 1 2 u 2 2 sin 2 a)i 

 (yho 2 — u ± 2 u 2 2 sin 2 a)* ^ v 2 — u^ cos a — (Vto 2 — u x 2 u 2 2 sin 2 a)* 



It is easily seen that the values of W found in § 4 for a = 

 and for a = ir may be deduced from the general expression 

 for W. 



It is convenient to write W/Uo in the form 



W 1, 1 + Z 



where 



7— (dV- i^i^sin 2 ^ 



Li r « 



v z — iiiii 2 COS a 



The expression for W/U has no meaning when Z, which 

 is positive, is greater than unity. When Z is less than unity, 

 we have 



w rz 2 z± z 6 "\ 



U L3 o 7 J 



Since all the terms of this series are positive, W/U 

 increases as Z increases. 



If we write u 2 -\-n 2 — 2w^ 2 cos a for iu 2 , we find 



z ^ 1 _ (*'-VX« 8 -«» 3 ) 



(V — W^COSa) 2 



Hence Z increases as a increases from to 7r. Thus for 

 given values of u x and u 2 , the radiated energy, W, increases 

 as the angle between the directions of u x and u 2 increases, 

 since both «j and it* are less than v. 



For given values of Vi and u 2 , the maximum value of Z 

 occurs when u = ir. Then we have 



7 __v(«i + w 2 ) i («— Wi)( w — Ws) 



Zjtt — = j_ __ 



and thus Ztt is positive and less than unity. Collecting these 

 results we see that for all values of a and for all permissible 

 values of u x aud u 2 , Z is positive and less than unity. 



§ 12. Since the loss of energy arises from the change in 

 the velocity of the sphere, it will be convenient to express W 

 as directly as possible in terms of w, the change of velocity. 

 As we require two other quantities to fix the circumstances 

 of the problem, we take wand| where 2u is the resultant 

 of Ui and u 2 and yfr is the angle between the positive 



