364 
PROFESSOR M. J. M. HILL ON THE MOTION OF FLUID, PART 
Suppose that in any of these cases any particular integral of the equation in X is 
taken. 
It is shown that the components of the velocity can be expressed in terms of X and 
differential coefficients of X, and that the current function is also known. 
Tn the case of a fluid, part of which is moving rotationally and part irrotationally, 
the boundary surface separating the rotationally moving fluid from that which is 
moving irrotationally contains the same vortex lines, and may be taken at the surface 
X=0. 
Now, if the integral taken of the equation in X do actually correspond to a case of 
fluid motion in which part of the fluid is moving rotationally and part irrotationally, 
the most obvious way to find the irrotational motion will be to find its current function 
from the conditions supplied by the fact that the components of the velocity are con¬ 
tinuous at the surface X=0. Examples I. and III. of this paper have been solved in 
this manner. 
If after taking any integral of the equation in X it be found theoretically impossible 
to determine the current function of an irrotational motion outside the surface X=0, 
which shall be continuous with the rotational motion inside it, then the integral in 
question does not correspond to such a case of fluid motion. 
In this method no assumption is made as to the distribution of the vortex lines (as 
in the method of Helmholtz) before commencing the determination of the irrotational 
motion. 
If, however, the rotational motion be known, the components of the velocity are 
known for this part of the fluid. Let the components of the velocity be expressed in 
Clebsch’s forms, so that y, X, xjj are known. 
Moreover, let the forms be so arranged that the surface separating the rotationally 
moving fluid from that which is moving irrotationally is the surface X=0. 
Then at this surface the components of the velocity are —• 
1 J dx dy dz 
Now, obtain in any manner a velocity potential (j> for space outside X=0 continuous 
with y all over the surface X = 0. This is theoretically possible always. 
If the velocity potential so obtained make the velocity and pressure continuous all 
over the surface X=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 X=0 are that there X — In order that the pressure may 
dx dx dy dy dz' dz 
be also continuous, it is further necessary that all over the surface X=0. 
The most obvious way of obtaining the velocity potential will be to apply Helm¬ 
holtz’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 
