260 Wisconsin Academy oj Sciences^ Arts, and Letters. 



The motion of these vortices, though, as we have shown, it does 

 not sensibly affect the visible motions of large bodies, may be such 

 as to affect that vibratory motion on which the propagation of 

 light, according to the undulatory theory, depends. The dis- 

 placements of the medium, during the propagation of light, will 

 produce a disturbance of the vortices, and the vortices, when so 

 disturbed, may re-act on the medium so as to affect the mode of 

 propagation of the ray. 



It is impossible, in our present state of ignorance as to the na- 

 ture of the vortices, to assign the form of the law which connects 

 the displacement of the medium with the variation of the vortices. 

 We shall therefore assume that the variation of the vortices, 

 caused by the displacement of the medium, is subject to the same 

 conditions which Helmholtz, in his great memoir on Yortex-mo- 

 tion, has shown to regulate the variation of the vortices of a per- 

 fect liquid. 



Helmholtz's law may be stated as 

 follows: — Let P and Q be two neigh- 

 boring particles in the axis of a vortex, 

 then, if in consequence of the motion 

 of the fluid these particles arrive at 

 the points P' ^, the line P' Q will rep- 

 resent the new direction of the axis of the vortex, and its strens^th 

 will be altered in the ratio of P' Q' to P Q. 



Hence if a, /?, y denote the components of the strength of the 

 vortex, and if f, tj, ^ denote the displacements of the medium, the 

 value of a will become 



d^ 



^ 



1 d^ , n d^ , c/f 



■■ai- a — . -f i3 — + J' — 



dx dy dz 



dx dy dz 



(1) 



J =y+a-^ + ^^+r-±- 

 dx dy dz 



We now assume that the same condition is satisfied during the 

 small displacements of a medium in which a, /?, y represent, not 



