280 Prof. M. J. M. Hill. Motion of Fluid, part of [Feb. 7, 



the rotationally moving fluid from that which is moving irrotationally 

 is the surface \=0. 



Then at this surface the components of the velocity are ^ 



dx dy dz 



Now, obtain in any manner a velocity potential for space outside 

 \=0 continuous with % all over the surface X=0. This is theoretically 

 possible always. 



If the velocity potential so obtained make the velocity and pres- 

 sure continuous all over the surface \=0, then a possible case of 

 motion will have been obtained. 



The conditions to be satisfied in order that the velocity may be 



continuous at the surface \=0 are that there ^ = 



ax dx ay ay dz dz 



In order that the pressure may be also continuous, it is further 



necessary that < ^K= C ^ all over the surface \=0. 

 J dp dt 



The most obvious way of obtaining the velocity potential will be to 

 apply Helmholtz's method of finding the components of the velocity in 

 terms of the supposed distribution of magnetic matter throughout the 

 space occupied by the rotationally moving fluid. 



It must, however, be remembered, as is remarked by Mr. Hicks in 

 his report to the British Association on "Recent Progress in Hydro- 

 dynamics," Part I,* " That the results refer to the cyclic motion of 

 the fluid as determined by the supposed distribution of magnetic 

 matter, and do not give the most general motion possible." It 

 appears also from Examples I and III of this paper that it is not possible 

 to assume arbitrarily the distribution of vortex lines, even when it 

 can be shown that the equations of motion are satisfied at all points 

 where the fluid is moving rotationally, and then to proceed to cal- 

 culate the irrotational motion by means of the supposed distribution 

 of magnetic matter. For in these examples, values of the components of 

 the velocity of a rotational motion, satisfying the equations of motion 

 throughout a finite portion of the plane of x, y, are found. Thus the 

 distribution of vortex lines, and, therefore, that of the supposed 

 magnetic matter over a finite portion of the plane of a;, y is known. The 

 surfaces that always contain the same vortex filaments are found. 

 Inside one of these the supposed magnetic matter is distributed, the 

 current function at external points is calculated by Helmholtz's 

 method, and it is shown that the velocity so found is not continuous 

 with the velocity at the surface, which separates the rotationally 

 moving liquid from that moving irrotationally. 



Another way (suggested by Clebsch's forms) of obtaining the 

 velocity potential will be as follows : — 



* Eeport for 1881, Part I, p. 64. 



