the Theory of -Electrons'. 275 



the equilibrium or want of equilibrium of the forcives of 

 which we should then have to take cognizance, instead of 

 finding, as in the theory of electrons, that the play of forcives, 

 in terms of which we describe the state of equilibrium of the 

 aether under the influence of fixed electrons or conductors, is 

 inadequate to account for what happens to the aether which 

 lies in the path of a moving electron. 



9. Some analysis, slightly modified from that given by 

 Larmor*, bearing on the subject under discussion is here sub- 

 joined. Suppose the medium compressible, incompressibility 

 bsing regarded as a limiting case, and let the surface bound- 

 ing- an electron-nucleus be considered as a surface in crossing; 

 which there is discontinuity of elastic quality, compressive or 

 rotational, or both. As the mechanical forces whose genesis 

 is in question are proportional to the squares of the rotations, 

 it is requisite that the equations should be correct as far as 

 terms of that order in those strains, though not necessarily in 

 the compressions. Let V denote the volume, and p the 

 density of an element of aether, which when free from strain 

 occupies volume V and is of density p , and let the pressure 

 be A 2 A, where A denotes the compression, i. e. (V — V)/V 

 or (p—p )/p; then the compressive potential energy is 

 A*A 2 /2p per unit mass. It may plausibly be supposed that 

 compression of an element does not afreet the rotational 

 potential energy, and accordingly the latter is represented by 



h a %f* + 9* + 1*?) P er un if mass instead of by iPo a2 (/ 2 + # 2 + ^ 2 ) 

 per unit volume, where (/, g, h) is the rotational strain. Let 

 £, t), £ be the components of the displacement of the element 

 of aether which is at the point x, y, z at the time t from the 

 position it occupied at the time zero. The conditions of equi- 

 librium are then to be derived from the variation of the 

 potential-energy function 



W = I L 2 (/' 2 + ( f + A 2 ) dm + — f A*A 2 rfm, 



wherein dm denotes an element of mass. Let D denote a 

 variation following the motion of an element. Then 



J dy d.z 



and 



t D A=-^--£(PB-£cD,)-!(DO. 



On conducting the variation for the portion of the medium 

 * Phil. Trans. 1894, A. pp. 747, 793. 



