44 Mr. G. F. C. Searle on the Impulsive 



charged sphere is impulsively changed by an infinitesimal, 

 amount, the change o£ velocity making any angle with the 

 initial velocity. In the present paper I complete the inves- 

 tigation by calculating the energy and momentum which are 

 radiated when the velocity of the sphere is impulsively 

 changed in any manner, the only restrictions being that both 

 the initial and the final velocities are less than that of light, 

 and that the sphere has no angular velocity about any axis 

 through its centre. The results in the restricted case, in 

 which the centre o£ the sphere moves along a single straight 

 line throughout, have been given by Heaviside and by- 

 Paul Hertz. 



§ 2. The investigation turns mainly upon a certain in- 

 tegral. In the evaluation of this integral I have been greatly 

 aided by Mr. G. T. Bennett, Fellow of Emmanuel College. 

 I had carried out the integration in the manner given in 

 § 10 below, but I was unable to put the result into a sym- 

 metrical form. Mr. Bennett then devised the very elegant 

 semi-geometrical process which is given in §§6, 7, 8, and, 

 guided by the result, reduced the complicated expression 

 arising in my method of integration to a symmetrical form.. 

 I am also indebted to Mr. F. J. W. Whipple, of the Merchant 

 Taylors' School, for verifying the integration by an alternative 

 method. 



§ 3. Now let us consider a sphere of radius a, carrying a 

 charge Q on its surface, and let us suppose that its velocity 

 is impulsively changed from Ux to u 2 , and that the angle 

 between u x and u 2 is a, heavy type denoting vectors. Then 

 it will be seen from § 31 of my earlier paper that, when the 

 pulse generated by the change of velocity has travelled out 

 through a distance r, which is very great compared with the 

 diameter of the sphere, the electric force in the pulse is at; 

 right angles to the radius and is given by 



Q | R(u 2 R)-u 2 BfojQ-Ti J 

 2Kral v-u 2 R u-u,R J' * * { } 



Here R denotes, for the moment, a unit vector along the 

 radius, and i^R denotes the scalar product of u, and R, while 

 K is the specific inductive capacity and v the velocity of 

 light. 



The magnetic force in the pulse is at right angles to both 

 E and R, and is given by 



H=eKVBE, (2) 



where VRE is a vector product. 



