JEther and Electrons. 699 



of work demanded being precisely zero : so that the strain- 

 figure is free to move through the medium *. 



10. It will now be convenient to classify strain-distributions 

 under three heads. Any distribution which we conceive as 



* So long as only an infinitely slow displacement of tlie strain-figure 

 through the medium is in question, complete mobility appears to "be 

 assured ; but in order that the strain-figure may be able to more through 

 the medium with finite velocity, and without involving dissipation of 

 energy, a further limitation seems called for. The need for this was 

 made evident by an objection expressed to me in general terras by 

 Professor Larmor, whcse view I have attempted to deal with in this 

 note. 



Let J), D', . . . ., functions of x, y, z, serve to define the distribution of 

 displacement, concerning which nothing is known, except perhaps that it 

 is not bodily translational displacement at each point of the medium. 

 Then, in accordance with the assumed continuity of the process by winch 

 a strain-figure is ideally producible, D, D', . . . . are continuous functions 

 of x, y, Z' } in other words, 



(a/a*, a/to a/a*o (i>, i>', . . . .) 



are all finite. But when the strain-figure is moving with velocity- 

 components II, V, TV. which are small compared with the velocity of 

 radiation, we readily find that 



while 



§ = -u|5_v|P-w|P i e te .=etc. ; 



o^ d-f cty dz 



d»D__<flT dD dV ^B_dW &D 

 dt 2 dt ' &x dt ' dy dt ' d~ 



+ (ui+vi+w|) 2 D 



\ d* di/ &/ 



etc. = etc. 



Though, strictly speaking, the differentiation refers to a point fixed in 

 space, dD/df, • • • • are (™ a generalized sense) of the nature of velocities, 

 5 2 D ~dt 2 , .... being of the nature of accelerations : and while the 

 assumptions already explained make dD/d£ everywhere finite for mode- 

 rate values of U, V, W, we shall not be able to avoid infinite values 

 of <)D 2 fot 2 , .... without introducing the further assumption that 

 d 2 D/d.r 2 , .... are every where finite. We must assume, in fact, that 



(a/a*, a/ay, a/a«o (d, d', . . . o 



are not only finite but continuous functions of the coordinates. If 

 impulsive time-changes of the (generalized) velocities ^D ; Q£, .... were 

 to accompany a finitely rapid motion of the strain-figure, dissipation of 

 energy would presumably be involved ; but it does not appear to m? 

 that a (finite and) continuous distribution of the strain-magnitudes 



(a/a*, a/to a/a2)(D,D',...o 



need be incompatible with the above-postulated constitution of a strain- 

 figure. Such continuity, in addition to the assumptions originally made, 

 would ensure perfectly free mobility without any loss of energy except 

 through radiation ; this latter being insignificant so long as the velocity 

 (U, V, W) is very small compared with the velocity of radiation, with a 

 corresponding restriction as to the acceleration d/dt (U, V, TV). 



