Con WAV — The Dynamics of a Rigid Electron. 173 



This term, therefore, together with the first term of the vector potential, 

 will give the first approximation.* 



Performing the differentiations, and putting the suffix 2 on p, we find for 

 the mechanical force exerted by Ci at pi on d at p^ 



e^e, { - lp^T-\p, - p.) + i (p3 - pi) /S'pi(p2 - pi)T-\pi - p^) 



-¥^KP2 - P^)T-\p. - P2) - |(P2 - />.)^Vi(p3 - Pi)T-Hp: - p.) (8) 

 + Vp,Vp,(p,-pOT'Xp,-pS 



This may be denoted by ^12. We shall denote - S^^ i.e. - 2(^12 + ^21) by A 

 and - 2 Vp^^s i.e. - S r(p2^i2 + pi^2i) by ;x. 



3. Eesults of Fiest Approximation. 



In applying these results to a rigid system, X and p. are calculated by 

 a process of summation which will extend to every pair of particles. In 

 the case of a continuous body the summation is replaced by a double volume 

 or sextuple integration. 



In either process certain terms will be found to disappear. Thus in 

 ^12 + S21 we have a term 



- f (|02 - Pl)'^Vl(P2 - pi)T-^ipi - P2) - |(P1 - p-)S^P2{pi - pi)T-^{pi - ps). 



This contains a scalar factor S{pi - p,) (pi - p^) which is zero since T{pi - p,) is 

 constant. For the same reason a similar term will disappear from 



F(p2|i2 + pi^2l). 



Also 



Vp^Vpiipo - pi) + VpiVp^ipi - pz) = V{pi - p2)Vp2pi. 



Let r denote a vector drawn from the origin to any point in the body 

 which we may term the hase point, and let o-i, 0-2, .. . be vector drawn 

 from the base point to various particles so that 



pi = T + (7i, 



P2 = r + 0-2, etc. 

 We have then since Ta^, Ta^ . . . are constant 



pi =^ 7" + F(0C7i 



Pi = r + V(b(Ti + Vio Vioa, etc. 



*The ictarded potential appears to have been first used by Heariside and Levi-Civita. It 

 was discovered about the same time independently by the author (Proceedings of the London 

 Mathematical Society, Series 2, vol. i.). 



