474 ON PHYSICAL LINES OF FORCE. 



so that the energy of the vortices in unit of volume is 

 and that of a vortex whose volume is V is 





(46). 



In order to produce or destroy this energy, work must be expended on, 

 or received from, the vortex, either by the tangential action of the layer of 

 particles in contact with it, or by change of form in the vortex. We shall first 

 investigate the tangential action between the vortices and the layer of particles 

 in contact with them. 



PROP. VII. To find the energy spent upon a vortex in unit of tune by 

 the layer of particles which surrounds it. 



Let P, Q, R be the forces acting on unity of the particles in the three 

 co-ordinate directions, these quantities being functions of x, y, and z. Since 

 each particle touches two vortices at the extremities of a diameter, the reaction 

 of the particle on the vortices will be equally divided, and will be 



-\P, -iQ, -\R 



on each vortex for unity of the particles; but since the superficial density of 



the particles is - - (see equation (34) ), the forces on unit of surface of a vortex 

 ZTT 



will be 



Now let dS be an element of the surface of a vortex. Let the direction-cosines 

 of the normal be I, ra, n. Let the co-ordinates of the element be x, y, z. Let 

 the component velocities of the surface be u, v, w. Then the work expended on 

 that element of surface will be 



(47). 



Let us begin with the first term, PudS. P may be written 



dP dP dP 



and u = n/3 my. 



