170 Proceedings of the Royal Irish Academy. 



considered by Abraham in great detail, and a result is obtained which is 

 a function of the velocity. A general expression was obtained by him, 

 and independently by Heaviside, for the energy radiated, and the former 

 also deduced * the " back-pressure " on the electron due to the radiation. 

 As these and similar results are continually quoted and made use of for 

 drawing deductions concerning the mass of electrons, and as it seemed 

 to the present writer that (with the exception of the expression for the 

 radiated energy) the results were of less extended scope than those who 

 made applications of them considered, it was thought advisable to 

 develop a different method for approaching them. A method of continued 

 approximation is explained hereafter, and the results of comparison with 

 the ordinary expressions are as follows : — No expression for the electromagnetic 

 mass of a spherical shell is onore correct than the simplest one ^e-ja, and the 

 general expression for the iacJc-jpresswe clue to radiation is not more correct than 

 the simpilest one ^e'v/u^. In fact, it is easy to see a p)riori that the general 

 results are not in general true. For example, consider the energy in a 

 given place of the medium due to a moving electrified sphere. We can 

 calculate the position in which the centre of the sphere must be, so 

 that the radiation from a certain part of the sphere will reach the 

 place at a certain time t. These positions will be different for different 

 parts of the sphere, and will occupy a certain length of the path of the 

 centre of the sphere, so that the energy will depend on the history of 

 the motion of the sphere rather than on the knowledge of the position 

 of the sphere at some one time. The only exception to this is when 

 destructive interference takes place, and this only happens for uniform 

 motion. For the more general case considered in this paper, it is found 

 for the first approximation that an electrified body of any shape moves 

 in the same way as a body in a liquid. The second approximation 

 shows that Lorentz's expression for the back-pressure is independent 

 of the shape of the system — a result noticed by Abraham, The third 

 approximation shows that Abraham's expressions for the longitudinal and 

 transverse mass are correct for the case of a sphere moving with uniform 

 acceleration, provided we neglect terms containing the square and higher 

 powers of the acceleration. The notation used is quaternion. 



2. Description of Method Employed. 



Let the masses of the particles placed at the corners of the rigid framework 

 be 7/^1, //(2 • • . . , and their electric charges c,, eg .... , and the vectors drawn 



* British Association Eeport, Cambridge, 1904- 



