148 Impulsive Motion of Electrified Systems. 



w = 2usin a, and the angle between w and u is -v/r=^7r — a. 

 Then, by § 31, when a is small, and therefore w is small, the 

 mean rate, R, of radiation of energy is 



p 2fiQ 2 u d (l — ?i 2 cos 2 a) sin a 

 3a?'(l— 7i 2 ) 2 

 This result is applicable until a. becomes so small that the 

 pulse arising from one impulse has not got clear of the sphere 

 before the next impulse occurs. The smallest permissible 

 value of sin a is au/(y— u)r, and in this case 



2/uQ 2 u l 1 — n 2 cos 2 a _ 1 — n 2 cos 2 a 

 ~_3vr 2 (l — n 2 ) 2 ' 1 — n 1 — n 



Here the radius of the sphere appears only through cos 2 «. 

 If we make a/r infinitesimal, we have cos a=l and 



R = (l + n)B,. _ 



Dr. Heaviside* has shown that R is the rate of radiation 

 of energy from a true point-charge moving with velocity u in 

 a circle of radius r. Hence we see that, however small the 

 sphere may be, the nfean rate of radiation when the successive 

 pulses just fail to overlap is greater than when they merge 

 into a continuous disturbance. 



§ 36. If a point-charge q has an acceleration f,the change of 

 velocity in time dt isfdt. The disturbance arising during this 

 time is confined to a shell contained between two spheres, one of 

 radius vt described about the position of q at the beginning 

 of the interval, and the other of radius v(t — dt) described 

 about the position of q at the end of the interval. The thick- 

 ness of the shell is thus (y— ucosd) dt or vh dt and the flux 

 of displacement K<?/l7r is distributed through this thickness. 

 Hence, putting f dt for w in (50) and allowing for the thick- 

 ness of the shell, we find that the electric force at a great 

 distance from an accelerated point-charge is 



B= J T f + (>r 1 -u)/cos 7 -j 



Kr L (v — u cos 6) 2 ' (v—u cos 0) 3 J 

 where 6 and y are the angles made by r with u and f. This 

 result agrees with that given by Dr. Heaviside f, who has 

 also calculated the energy and the momentum carried away 

 in the expanding shell. 



§ 37. In the preparation of this paper I have been greatly 

 assisted by Mr. D. C. Jones of Pembroke College. He has 

 not only written out the manuscript for the press but has 

 also verified all the formula?. My thanks are also due to 

 Prof. J. J. Thomson for reading the paper in manuscript, 

 and to Dr. A. H. Bucherer of Bonn '-for reading the proofs. 



Cavendish Laboratory, Cambridge. 



* ' Nature,' Nov. 6, 1902, equation (10). 



t 'Nature/ Nov. 6, 1902. See also M. Abraham, Theorie der Elek- 

 trizitat, ii. p. 97. 



