618 EEPORT— 1895. 



which a change of velocity accompanies any continuous change of direction, this 

 change of velocity is always a very smooth one, and not abrupt. 



Alter remarking that if the velocity of a ray in any given direction were 

 dependent equally on the resistances oflered to ether-movement in every direction, 

 this velocity would in all cases be entirely independent of any particular direction 

 or directions in the structure, which would in all cases be isotropic, he says that 

 the experimental facts show that the truth lies between the two extremes indi- 

 cated ; that the velocity of a ray depends neither on all the resistances to ether- 

 movement experienced in all directions taken equally, nor on the resistance 

 experienced in a single solitary direction, but depends equally, or almost equally, 

 on the resista7ices afforded in all the directions included within some wide limits of 

 angular inclination, this being the only kind of relation which would be in 

 harmony with the great smoothness of the change of velocity presented when a 

 continuous change of direction is made. 



He then suggests that the simplest sort of relation which the velocity can be 

 conceived to bear to the resistances offered by the structure to ether-movement is 

 for the resistance whose direction is that of the polarisation of the ray — i.e., the 

 direction in which the algebraically deduced wave-vibration takes place — to exert 

 a maximura influence, and the etiect of the resistances in directions inclined to 

 this to diminish as the inclination increases, the decrement of influence for direc- 

 tions near the direction which furnishes the maximum eflect being, however, very 

 small indeed. 



He points out that if this simple kind of relation obtains, the velocity Jigure— 

 i.e., the figure whose radii express the different velocities proper to diflerent direc- 

 tions of polarisation for rays traversing a crystal — must exhibit a smoothed cur- 

 vature derived indeed, but having a very different aspect, from that of the 

 corrugated surface whose radii would express the relative facility of ether- 

 movement taking place in different directions in the same crystal ; and that the 

 simplest conceivable result of such a smoothing or averaging will be for the 

 velocity-figure to approximate as closely as we please to the result obtained by 

 treating the vebcity appropriate to any direction of polarisation whatever as the 

 resultant of three components acting in some particular three widely separated 

 directions, each component, in harmony with the averaging referred to above, 

 being greater or less as the direction of the resultant which is being resolved lies 

 nearer to or further from its direction, and being zero when the resultant lies in 

 the plane of the remaining two components. The relative lie of the three direc- 

 tions will, of course, depend on the nature of the crystal structure. The reason 

 for taking three directions is that this is the least number which can be employed 

 consistently with generality. 



He proceeds to show that the simplest figure thus obtainable is an ellipsoid, of 

 which the three axes are conjugate diameters, and calls attention to the fact that 

 the number of the sets of three axes which will fulfil the requisite conditions in 

 any given case is unlimited. 



From the fact that the veloritj'-figure is in all crystals found to be an ellipsoid 

 (specialised, indeed, in some of the crystal systems), he finally argues that the 

 velocity of a ray is an average effect of the different resistances to ether-movement 

 <ffered in different directions of the nature above e.rplained ; and that the combi- 

 nation or averaging by which so simple a figure as the ellipsoid is reached must 

 not only extend over a wide range of resistances for each velocity, but also that it 

 must be so nearly uniform in its application throughoid some considerable j)ortion of 

 this range as to preclude entirely all merely local effects of the structural features 

 of the crystal on the contour of the velocity -figure. 



In closing, the writer remarks that the directions which give maximura or 

 minimum velocity— 1.(?., those of the principal axes of the ellipsoid— will not 

 necessarily be directions of maximum or minimum facility of ether-movement, the 

 indents and protuberances of the corrugated figure whose radii express the relative 

 facility of ether-movement in diflerent directions not being traceable as such on 

 the velocity-figure. 



Also that tue directions of the principal axes of the velocity-ellipsoid will not 



