tT 



364 REPORT— 1893. 



rotational aether. And here there is a point which demands explanation. 

 The energy-function is the same in both the labile aether and this 

 rotational aether ; but the boundary conditions are difiFerent, being in the 

 one case those of the elasticity of solids and in the other those of pure 

 rotational elasticity. Yet, in the treatment of the subject proposed here, 

 emphasis is laid, after MacCullagh, on the fact that the energy-function 

 implicitly involves in itself the boundary conditions. This difficulty is 

 •elucidated by observing that the expression 



1 f r fdc_dvy fdi_d!:y fd,_cu\n 



2j lUif/ dzj ^U dxj ^{dx dyj / '' ' 



given by Lord Kelvin ^ for the potential energy of the labile aether does 

 not represent the localisation of the energy, considered as that of an 

 elastic soHd. It is in fact derived from the appropriate expression for an 

 ■elastic solid 



2 J L\dy dzJ \dz dxJ \dx dyJ 



__^rdr,di:^d^dl^dldrr^ 

 \dy dz dz dx dx dyJ J 



by integration of the second term by parts ; and at an interface between 

 different media, a surface term which will be found to be the difference 

 for the two media of the values of the expression 



j={<f-f)-(f-f)-(f-f)}"« 



■is thus thrown away. Now a superficial distribution of energy is repre- 

 sented mechanically by a surface-tension of equal intensity ; so that a 

 surface-tension of this amount, varying from point to point, assists in 

 keeping up the equilibrium of the interfacial layer, in addition to the 

 surface forces indicated by MacCullagh's analysis. 



Elucidation of a General Dynamical Principle. 



24. A cardinal point in this correlation of different theories is the insist- 

 ence on the validity of the proper application of MacCullagh's doctrine that 

 the energy-function of a medium, provided it is correctly localised, con- 

 tains implicitly in it the aggregate of the boundary conditions at an inter- 

 face between two different media; and that notwithstanding any apparent 

 discrepancy in continuity that may still be outstanding after the conditions 

 so obtained have been appHed to the problem. The same principle had 

 previously been formulated by Green,- in similar terms ; ' one of the 

 advantages of this method, of great importance, is, that we are neces- 

 sarily led by the mere process of the calculation, and with little care on 

 our part, to all the equations and conditions which are requisite and 

 sufficient for the complete solution of any problem to which it may be 

 applied.' On the practical application of this procedure some fresh 



1 Lord Kelvin (Sir W. Thomson), PMl Mag., 1888. 



- George Green, ' On the Laws of the Keflexion and Eefraction of Light,' Trans. 

 €amb. Phil. Soc, December 11, 1837; Math. Papers, p. 246. 



