Discontinuity in a Rotationally Elastic Medium. GO 7 

 Now we may introduce a new vector (£, w, f) defined by 



(a, fi, y) = | (ft „, f) j J (X, T, Z) = Curl (6, ,, f ) . . (8) 



In a rotationally elastic aether, (£, ??, f) is interpreted as 

 linear displacement ; the constants of inertia and elasticity 

 of the medium must be supposed very large so that all the 

 motions and displacements are small and can be analysed into 

 independent differential strains and rotations*. Otherwise, 

 we may begin with such a medium and take the energy 

 functions to be 



T=*Aj(f« + # +'(*)<*!■; 



where (/, g, h) = Curl (f, 77, J). 



The principle of least action gives the equations of motion ; 

 and comparing the potential energy function W with electrical 

 energy we obtain the correspondence given in (8). 



Regarding then the aether as an incompressible, rotationally 

 elastic medium, its dynamical equations are 



V 2 (ft^,?)=J|^ (ft v,Q, .... (9) 

 combined with 



M + !? + ff=o (io) 



O^ Oi/ 02 



Now in order to determine completely the motion in the 

 medium, we must know in addition the conditions at the 

 limiting surface of the medium. For instance, suppose we 

 know the motion of the boundary at each instant, and thus 

 the acceleration of each point ; then this set of values for the 

 acceleration may differ at any moment from those obtained 

 from the internal equations of motion of the medium. 

 Consequently a discontinuity of the second order is set up at 

 the boundary and is propagated into the medium. Again, if 

 the velocity or pressure at the boundary be suddenly altered, 

 a discontinuity of the first order will be set up. 



In electrical terms, we may be given a boundary in the 

 aether at which the electric or magnetic force is assigned ; 

 then there may be set up a discontinuity of the first order 

 which is propagated into the aether as a wave-front at which 

 the electric and magnetic forces are not continuous. Or, 

 again, if the electric current is assigned at the boundary, 

 * Larmor, ' Mther and Matter,' p. 332, 



