﻿522 The Hon. J. W. Strutt on Double Refraction. 



body, and that in a manner dependent on the direction of vibra- 

 tion. An extreme case will make this evident. Suppose that 

 the ellipsoid degenerates into a circular disk of inconsiderable 

 thickness. It is clear that if the vibration be performed in a 

 line perpendicular to the disk, the fluid, which cannot readily 

 pass from the one side to the other, will greatly impede the 

 motion— that is, increase its period; for there is no question 

 here of a loss of energy from friction or viscosity. 



It is equally evident that if the motion be in the plane of the 

 disk, the fluid has no effect and might as well not be there. We 

 see, then, that,*to all intents and purposes, the disk has a density, 

 or rather inertia, of variable magnitude dependent on the line of 

 vibration, and symmetrical round an axis, and are reminded of a 

 uniaxal crystal. Next suppose that we try to make the disk vi- 

 brate in a line oblique to itself. It would at once appear that 

 such a vibration cannot be performed without an additional con- 

 straint, which we may suppose applied. The system would then 

 perform pendulous vibrations whose period is a function of the 

 position of the line in which the centre of inertia is made to 

 move. Lagrange's general method leads immediately to a solu- 

 tion of the whole problem : — 



T= kinetic energy = \ {Pa> + Q(y« + z*)} 3 

 V= potential energy = £ (a? 2 + y* + z*)', 

 whence the equations of vibratory motion, 

 Pa? +/a#=0, 



showing that vibrations along x cannot be performed synchro- 

 nously with vibrations along y or z. This is on the supposition 

 that the body is free ; but if it be constrained to a line f making 

 an angle 6 with a?, we have 



T=if(Pcos 2 + Qsin 2 0), 



whence 



(Pcos 2 <9+Qsin 2 0)f+/^=O, 



so that the period t is 



2 / Pcos 2 <9 + Qsin 2 <9 



V fi 



V 2 * ' 



