Elastic Solid yEtlier. 557 



Our bodily equation, by a straightforward application of the 

 dynamical principle h f L^ = to the MacCullagh form (8),. 

 is found to be 



£=-W(AV7i?)-V(&Sv?). . . . (11) 



Operating by SV( ) an d by VV( ), we find (as we may 

 anticipate from the usual treatment after Green) that the 

 convergence and curl behave quite independently of one 

 another. In tbe present theory, and this is where the present 

 theory differs from Green's, this statement is true of tbe 

 interface region as well as of the other two regions, the fret 1 

 Aether and the crystal. 



Thus in the present theory ice are not confronted ivith the 

 appearance of a wave of condensation when a wave of curl 

 suffers reflexion. 



Optical interpretations are concerned only with the curl VV 7 ?- 

 We may suppose initially, and therefore by (11) permanently, 



that SV V = = SV^; or, what is mathematically equivalent, 

 we may henceforth suppose w to stand for that part only of the 

 disturbed displacement which has curl and not convergence. 

 The bodily equation becomes 



;;=_vv(/«vv;), (12) 



mid the potential energy is confined to the cubic term. 



(12) is MacCullagh's equation and (11) is Green's. (12) is 

 also Maxwell's bodily equation if we identify tj with the 

 magnetic force and h with the reciprocal of the permittivity. 

 The condition that 77 is continuous at an interface is common 

 to all four theories, MacCullagh's, Green's, Maxwell's, and 

 the present one. MacCullagh's and Maxwell's second boundary 

 condition is that hY\Jrj is tangentially continuous, whereas 

 Green's is different ; and to-day it is well known that 

 MacCullagh's and Maxwell's condition is in harmony and 

 Green's condition is not in harmony with the optical facts of 

 reflexion at the boundaries of transparent bodies. "What does 

 the present theory say on this second boundary condition ? 



The following appears to me to prove that the present 

 theory gives the desired optical condition. 



Examining the sense in which It is large at an interface 

 the following at once appears. Let the axis of z be taken 

 perpendicular to the interface, h is given by the six second 

 space derivatives of a, 



ffrx, I'yih 9xy ; ffxz, <Jyz ', <Jzz- 



Of these, in the interface region, the first three g xx , g yy , a,.„ 



