J. W. Gibbs — Elastic and Electrical Theories of Light. 473 



cules and inter molecular spaces, and the extent to which the 

 molecules take part in the motion are changed. There are 

 cases in which these vary rapidly with the period, viz : cases 

 of selection absorption and abnormal dispersion. But we may 

 fairly expect that there will be many cases in which the char- 

 acter of the motion in these respects will not vary much with 



C f 



the period. -^ and -^ will then be sensibly constant and we 



have an approximate expression for the general law of disper- 

 sion, which agrees remarkably well with experiment.* 



If we now return to the equation of energies obtained from 

 the elastic theory, we see at once that it does not suggest any 

 such relation as experiment has indicated, either between the 

 wave-velocity and the direction of displacement, or between 

 the wave-velocity and the period. It remains to be seen 

 whether it can be brought to agree with experiment by any 

 hypotheses not too violent. 



In order that V 2 may be a quadratic function of any set of 

 direction-cosines, it is necessary that A D and & D shall be inde- 

 pendent of the direction of the displacement, in other words, 

 in the case of a crystal like Iceland spar, that the direct action 

 •of the ponderable molecules upon the ether, shall affect both 

 the kinetic and the potential energy in the same way, whether 

 the displacement take place in the direction of the optic axis 

 or at right angles to it. This is contrary to everything which 

 we should expect. If, nevertheless, we make this supposition, 

 it remains to consider B ND . This must be a quadratic function 

 of a certain direction, which is almost certainly that of the dis- 

 placement. If the medium is free from external stress (other 

 than hydrostatic), B ND , as we have seen, is symmetrical with 

 respect to the wave-normal and the direction of displacement, 

 and a quadratic function of the direction-cosines of each. The 

 only single direction of which it can be a function is the com- 

 mon perpendicular to these two directions. If the wave- 

 normal and the displacement are perpendicular, the direction- 

 cosines of the common perpendicular to both will be linear 

 functions of the direction-cosines of each, and a quadratic 

 function of the direction-cosines of the common perpendicular 

 will be a quadratic function of the direction-cosines of each. 

 We may thus reconcile the theory with the law of double re- 

 fraction, in a certain sense, by supposing that A D and 5 D are 

 independent of the direction of displacement, and that B^ and 

 therefore V 2 is a quadratic function of the direction-cosines of 

 the common perpendicular to the wave-normal and the dis- 



* This will appear most distinctly if we consider that V divided by the velocity 

 of light in vacuo gives the reciprocal of the index of refraction, and p multiplied 

 by the same quantity gives the wave-length in vacuo. 



