246 Prof. J. W. Gibbs^s Comparison of the Electric Theory 



in our mathematical theory, even if we do not regard this as 

 expressing the actual facts with absolute accuracy. But the 

 case is not so simple with an elastic theory in which the forces 

 resisting certain kinds of motion vanish, so far, at least, as 

 they are proportional to the strains. The first requisite for 

 any sort of optical theory is that the forces shall be propor- 

 tional to the displacements. This is easily obtained in general 

 by supposing the displacements very small. But if the 

 resistance to one kind of distortion vanishes, there will be a 

 tendency for this kind of distortion to appear in some places 

 in an exaggerated form, and even to an infinite degree, how- 

 ever small the displacements may be in other parts of the 

 field. In the case before us, if we suppose the velocity of the 

 missing wave to be absolutely zero, there will be infinite con- 

 densations and rarefactions at a surface where ordinary waves 

 are reflected ; that is, a certain volume of asther will be con- 

 densed to a surface, and vice versa. This prevents any treat- 

 ment of the extreme case, which is at once simple and satis- 

 factory. The difficulty has been noticed by Sir William 

 Thomson, who observes that it may be avoided if we suppose 

 the displacements infinitely small in comparison with the 

 wave-length of the wave of compression. This implies a 

 finite velocity for that wave. A similar difficulty would pro- 

 bably be found to exist (in the extreme case) with regard to 

 the deformation of the asther by the molecules of ponderable 

 matter, as the fether oscillates among them. If the statical 

 resistance to irrotational motions is zero, it is not at all evident 

 that the statical forces evoked by the disturbance caused by 

 the molecules would be proportional to the motions. But this 

 difficulty would be obviated by the same hypothesis as the 

 first. 



These circumstances render the elastic theory somewhat 

 less convenient as a working hypothesis than the electric. 

 They do not necessarily involve any complication of the 

 equations of optics. For it may still be possible that this 

 velocity of the missing wave is so small, that the quantities on 

 which it depends may be set equal to zero in the equations 

 which represent the phenomena of optics. But the mental 

 processes by which we satisfy ourselves of the validity of our 

 results (if we do not work out the whole problem in the 

 general case of no assumption in regard to the velocity of the 

 missing wave) certainly involve conceptions of a higher 

 degree of difficulty on account of the circumstances mentioned. 

 Perhaps this ought not to affect our judgment with respect to 

 the question of the truth of the hypothesis. 



Although the two theories give laws of exactly the same 

 form for monochromatic light in the limiting case, their devia- 



