ON PHYSICAL LINES OF FORCE. 509 



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 motion in a medium 

 containing vortices, the vibrations being perpendicular to the direction of pro- 

 pagation. 



Let the waves be plane-waves propagated in the direction of z, and let 

 the axis of x and y be taken hi 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, tending to 

 move the part next the origin hi the direction of x. 



Let Y be the corresponding tangential stress in the 

 direction of y. 



Let &, and &, be the coefficients of elasticity with respect 

 to these two kinds of tangential stress ; then, if the medium 

 is at rest, 



X=kl dz' Y=k *dz- 



Now let us suppose vortices in the medium whose velocities are represented 

 as usual by the symbols a, /8, y, and let us suppose that the value of a 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 increasing 



at the rate pr -j- 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 = u.r -=- . 



4?r at 



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



,, , dy 1 da 



Y L J_ . _ iii - 



JL A/- j Utl 7 



dz 47r at 



. ., v , dx I dp 



Similarly, X = *, -j- + fir -j- 



