180 Proceedings of the Royal Irish Academy. 



If we perform the differentiations, we shall find that the total internal 

 forces will involve the velocities together with their first, second, and third 

 differential coefficients — terms which cannot arise from consideration of any 

 expression for electromagnetic mass which involves only the velocities, or 

 from any expression for radiated or " wasted " energy which involves only 

 the first and second differential coefficients of the velocity. 



If, therefore, these expressions fail at the third approximation, a fortiori, 

 they will fail at higher approximations which can be readily written down 

 by the methods we have explained. We shall consider then a special case, 

 that of a system in translational motion. On performing the differentiation 

 we find 



- S6T-\p - pdSi^,p\S(p - p,)p, - 18T-\p - p^)i,,'S{p - p)p\\, 



= ?,T-^{p-p;)[V{p-p,)p,J^^, 



where the terms not involving the accelerations are omitted, as they will 

 contribute nothing to the final result. We get finally for the resultant 

 of these forces an expression of the form 



T St (f)T + rSfTpT, 



where ^ and ip are self -conjugate linear functions. When the body is isotropic, 

 (f) and \p become constants, - ki and - kz, so that the force is 



— KitStt — JC'/Tt'^. 



If m is the mass of the electron, the total retarding force due to the first 

 and third approximation is 



- ( {m + kiT-) T + ki tStt). 

 Writing 



T = T Vt~^T + tSt'W 



the force becomes 



- ( {m + hr') T Vt~'t + (m + (h + h) f ^f/Sf-V). 



In other words, if we resolve the force along and at right angles to the 

 velocity, the coefficient of the former component is - (m + hr'^) and of the 

 latter - (m + [ki + kzjr^). These coefficients with reversed signs are termed 

 by Abraham the " longitudinal " and " transverse " masses respectively. 



