492 Mr. A. M c Aulay on Quaternions as a 



I sought and found the integral in attempting to consider, 

 from a fresh point of view, Sir "William Thomson's vortex-atom 

 theory and Prof. Hicks's proposed modification of it ; and 

 though I have not made any serious attempt to apply it in 

 this direction, I still think that it can be made of use in dis- 

 cussing how large groups of vortices act on other large groups. 



There would be apparent attractions or repulsions of a defi- 

 nite kind between vortices if the acceleration dajdt could be 

 expressed as a function of the vortices. In both the theories 

 just mentioned the fluid is assumed to be incompressible and 

 subject to no external forces, so that v=0. In Sir William 

 Thomson's theory the fluid is unbounded, and in Prof. Hicks's 

 it is infinite but bounded at certain places by what may be 

 called bubbles. In these bubbles the pressure is zero. We 

 have then from equation (32) for both cases 



4^= v {JJ tt ^f) + jp{V<rr-v("72i v «<fc}; (34) 



so that the vortices will move as though subject to a force 

 potential to given by 



4vw= -JJm^P - jj"j8^V<7T- V (^/2)}v»*. (35) 



I do not propose to discuss the bearing of these results here. 

 I merely give them to indicate that the integral just given 

 may, notwithstanding its apparent complexity when stated 

 perfectly generally, prove of great utility. 



A very different form may be given to the above integral. 

 It will probably hint to one familiar with quaternion methods 

 one way of proving the result. It is convenient to change 

 the notation. Instead of the former <r, Uv, ds, ds, and \7, 

 write now a' ' , UV, dsf, ds', and y'. Let p' , the vector coordi- 

 nate of any point, be supposed a function of an independent 

 variable vector p (say the coordinate of the point's initial 

 position), and t the time. Let u now stand for the reciprocal 

 of the distance between two points in the p space, instead of 

 as just now in the p' space. Similarly let Uv, ds, c/?, and v 

 refer to the p space. Finally, let a= — ViScr'/o/. Then 

 instead of equation (32) we may write 



4:tt(v + P) = jj 4 (i> H- P)SUi^ ds + J3J SvwvO ,2 /2)d9 



