Conway — The Dynamics of a Rigid Electron. Yll 



way find the usual expressions for the electromagnetic mass of a uniformly- 

 electrified spherical shell or sphere. A similar result holds for the mean 

 value of 01 ; in fact, the mean value of ^\{p) where the axes of ^i have 

 any directions is \in"p, Spcj^ip = - 1 is an ellipsoid, and the quantities 

 ffi> ffz, g^i are all included in the limits ^wf and \m" if all the charges 

 «!, ^2 . • . are of the same sign. For an elongated body the value of g will 

 usually he greatest for the axis which most nearly coincides with the 

 greatest length of the body. The function 03 is not self -conjugate. Its 

 invariant m" or - Si^ii is zero so that its mean value is zero. Unlike 

 01, its value depends on the base point. If we denote the values of the 

 functions at a base point / by <I>i, O2, ^z where $1 = 0i we obtain on 



putting T = T = a 



02 (p) = «I>2(/0 + ^i{Vpa) 



(f,,(jj) = $3(p) + Vci<P/p) + <l>'2(Fpa) + Vu^,{Vpaj. 



It may be noticed that these expressions leave unchanged in form the 

 expression for Kinetic Energy which is therefore, as it ought to be, 

 independent of the base-point. By properly choosing the base-point it is 

 possible to make O2 self- conjugate. If <l>2 is self -conjugate, then 



(92 - (j)\)p = <pi(Vpa) - Va(j)i(p). 



If (02 - 0'2)jo = '2VZp where ^ is the spin-vector of 02, then 

 2S(tZ,p = Sa(pi{Vpa) - Saatpip, 

 = >S'0io-pa + tS(T(jjipa, 

 = Sa\Pi( Vap), 



••■ - 2^ = ^,(«), 

 where i/^i is the auxiliary function of 0i. Hence a = - 2^r^(^j. Thus the 

 point T - 2^1"^ (^) is independent of the base-point. The above process 

 is exactly analogous to the simplification arising in ordinary dynamics 

 from choosing the centroid as base-point ; in fact, the functions xa, x'2 i^ 

 the last paragraph have no self-conjugate part, so that taking the centroid 

 as base-point makes them disappear. It is easy to see that 02 can be made 

 to disappear if the system has three places of symmetry. 



5. Hydeodynamical Analogy. 



The above results then form an addition to the analogies wdiich are 

 already known to exist between electrical theory and the dynamics of a 

 perfect fluid. There is however one difference to be noted. An impermeable 

 body immersed in a liquid will have an addition to its mass which, generally 



K. I. A. PF.OC, VOL. XXYII., SECT. A. [25] 



