applied to the Action of Magnetism on Polarized Light. 91 

 and the energy t , . j 



= -!-/^ 2 Vby Prop. VI.*, 

 whence 



A=^-firxV ......... (144) 



for the axis of x, with similar expressions for the other axes, V 

 being the volume, and r the radius of the vortex. 



Prop. XIX. — To determine the conditions of undulatory mo- 

 tion in a medium containing vortices, the vibrations being per- 

 pendicular to the direction of propagation. 

 ■. Let the waves be plane-waves propagated in the direction of z, 

 and let the axis of x and y be taken in the directions of greatest 

 and least elasticity in the plane xy. Let x and y represent the 

 displacement parallel to these axes, which will be the same 

 throughout the same wave-surface, and therefore we shall have x 

 and y functions of z and t only. 



Let X be the tangential stress on unit of area parallel to xy t 

 .tending to move the part next the origin in the direction of x. 



LetYbethe corresponding tangential 

 stress in the direction of y. 



Let k x and k 2 be the coefficients of 

 elasticity with respect to these two kinds 

 of tangential stress ; then, if the medium 

 is at rest, 



dx dy 



X ~ k 'Tz> X - k *dz 



Now let us suppose vortices in the medium whose velocities 

 are represented as usual by the symbols a, /5, 7, and let us sup- 

 pose that the value of u is increasing at the rate -j-, on account 



of the action of the tangential stresses alone, there being no 

 electromotive force in the field. The angular momentum in the 

 stratum whose area is unity, and thickness dz } is therefore in- 

 creasing at the rate — firj-dz; and if the part of the force Y 

 which produces this effect is Y', then the moment of Y' is —Y'dz, 



so that Y l = — - r -ixr-j-. 

 4iir r dt 



The complete value of Y when the vortices are in a state of 

 * Phil. Mag. April 1S61. 



