Impulsive Motion of Electrified Systems. 147 



Thus we obtain 



&**+'« **_ C* { n _ (1-rf) cos* J, - ^ 



Since cos # = (!—/«)//( and Aa.6d0=dhjn, this becomes 



i+tt 



Hence, finally, 



_ mQV- 3 sin -v/r cos -v/r \~2n(3 — *2n 2 ) _ 1 ^ np 1 + " ' 



4a 



i^cos -v/rr 2^(3 — 2»") _ o . i + » "[ 

 u/i 4 L 3(1— n 2 ) ° g l — ?i_]' 



By symmetry the momentum in the pulse has no component 

 perpendicular to the plane of u and w. 



§ 33. When n is small but, of course, not zero we find that 



where wig — 2/*Q 2 /3a, the electromagnetic mass o£ the sphere 

 for infinitesimal speeds. 



It is easily found that the whole momentum in the pulse 

 can be expressed in the vector form 



P=(w /10w 2 ){.3ti . r + w(uw) }. 



The angle (3 between P and u has its maximum value 

 when tan->|r = 2/y/3, the maximum value being given by 

 tanj8=(48)~*. These values correspond to ^ = 49° 6' and 

 £ = 8° 13'. 



§ 34. When n is nearly equal to unity, so that (1 — ?r) _1 is 

 large, we again obtain simple expressions for P x and P 2 . 

 Since log x is negligible in comparison with x when x is large, 

 we see, by § 32, that, when u is nearly equal to r, 



p _ mjv 2 (l — n 2 sin 2 yfr) p _ m Q w 2 sin "<jr cos yjr 

 1_ 2(1-« 2 )V '"' 2 ~~ 4(1— y?> 

 The resultant momentum in the pulse may be regarded as 

 being in the direction of u. For, if j3 be the angle between 

 P and u, we have 



Po (1 — ir) sin i/rcosi/r 



Inn H — — = — — 



ianp_ pi _ 2 (l-rc 2 sin 2 ^) > 



and tan/3 vanishes when yjr = and when yjr=^n: The 

 maximum value of tan/3 is the small quantity i(l — » 2 )^, and 

 this value occurs when tani^ = (l — n 2 )~*. 



§ 35. If the centre of the sphere be made to travel with 

 velocity u round a polygon inscribed in a circle of radius r, 

 and if 2x be the angle which each side subtends at the 

 centre, the vector change of velocity at each angle is 



L 2 



