178 Proceedings of the Roijal Irish Academy. 



speaking, will increase as the size of the IdocIj increases on account of a 

 larger body of liquid being disturbed. On the other hand, a given electrical 

 charge will have its inertia increased the smaller the space which it occupies. 

 The analogy therefore, although it directs our attention to the surrounding 

 aether as the true seat of the momentum and energy, leads us to think that 

 an electron has a more intimate connexion with the medium, forming in fact 

 according to the ideas of Larmor, some description of knot. 



6. Secoxd Approximatiox. 



Proceeding to the next approximation we have for the scalar and vector 

 potentials of the charge c^ at the point p^ 



CT = - U'^Ci'p^, 



so that this part of the force becomes 



-^u-Ho'p,. 



The whole system therefore exerts on any neighbouring particle e a force 

 whose value is 



If we take r such that t^Ci = ^Cipi, we get the force equal to ^ew^r. 

 Hence the internal reactions of the system in this case are equivalent to 

 a single force - |2<"^(2ei)^r acting through the centre of mean position of 

 the system for multiples ei, e^ . . . . This result is independent of the 

 form of the body. The activity of this force is 



Hence if a body moves in an almost periodic orbit, the energy wasted is 



— a well-known result. To understand better the part played by this force we 

 consider two examples of a simple nature. Let m denote the total " mass " 

 of a spherical electron, and let this force be denoted by kp where h is a 

 constant, and suppose that the force varies as the distance from the origin 

 being equal to - Ip. Then we have 



- Jc'p + m'p + Ip = 0. 



This or any linear equation of similar type 



A^p(") + hpi"-'^ + . . . = 



