Impulsive Motion of Electrified Systems. 131 



and (/>, where cf> is measured round the axis of motion. Then 



dQ 



1(g)* 



has the same value for the radius defined by ir — 6 and <£ + 7r 

 as for the radius defined by 6 and </>. Hence, if we expand 

 (1 — 7icos#) — 2 in the expression (35) for P and integrate 

 term by term, we see that the integral of the first term is 

 zero since, for a given value of dw, the parts of the pulse 

 defined by 6 and cj> and by ir — 6 and cf>-\-7r make equal and 

 opposite contributions to that term. Thus, going as far as 

 the second term, we find that for small values of n 



V=^^(^Jr l ^dc SeJa,<L. . (39) 



Hence, for small speeds, J? is proportional to n d r and there- 

 fore, by (21>), U-Uo is proportional to n\ By (13) it 

 follows that TT — T is proportional to n 4 . 



§ 19. Sphere with a uniform surface-charc/e. — This is, so far 

 as I know, the only system to which the method explained in 

 the present paper has been applied. The application was made 

 by Dr. Heaviside*. If Q be the charge and a the radius of the 

 sphere, dQ/dz = Q/2a, so that dQJdz is independent of z, and 

 thus the integrations with respect to z can be made at once. 

 We thus obtain, by (37). 



-r V _ /jlQ-h' 2 f" sin 3 '0 dO 

 ~ 4a J ~(T—n cos~6>) 2 ' 



where n=u/v. Putting 1 — n cos 6 = h, we find 



l+n 



■±<(u ! v jt Ir nr It rr } 



\-n 



2a \ti *v-u r 



_/xQ 2 ^/l „* i£ \ 



" a A3" 1 " 5t? a + 7^ + •••/' 



The electrostatic energy of the sphere at rest is given by 



IL= 



Q* p* Q 2 



2K 



The Electrician,' Nov. 29, 1901. p. 210. 

 K2 



