21G 
PROFESSOR M. J. M. HILL OX A SPHERICAL TORTEX. 
4. If c be not equal to a, then the surface containing the same particles, when the 
constant vanishes, breaks up into an evanescent cylinder and an ellipsoid of 
revolution. 
Now the velocity potential of an ellipsoid moving parallel to an axis is known. 
This velocity potential, with a suitable relation between k and Z, will make the 
normal velocity at the surface of the ellipsoid continuous with the normal velocity of 
the rotational motion inside the ellipsoid, but it does not make the pressm'e con¬ 
tinuous. Hence, if fluid can move outside the ellipsoid continuously with the 
rotational motion inside (described in section 1 abovm), then the motion outside the 
ellipsoid must be a rotational motion. 
5. It cannot be argued that the application of Helmholtz’s method to determine 
the whole motion from the distribution of vortices inside the ellipsoid must determine 
an irrotational motion outside the ellipsoid continuous with the rotational motion 
inside, because Helmholtz’s method determines the irrotational motion by means of 
the distribution of vortices only when that distribution is known throughout space. 
This is not the case in the problem under discussion. For here the rotationally 
moving liquid has been arbitrarily limited by rejecting all the vortices outside the 
ellipsoid, and it is not known beforehand that the rejection of these vortices is 
possible. 
6. Yet, on account of the interest of the problem, the paper contains a calculation of 
the velocity components in Helmholtz’s manner, supposing the only vortices to be 
those inside the ellipsoid, i.e., starting from the values of the velocity components 
n = - X [z — Z), 
2 /’ , 
= ,0 Z), 
V) = Z — (2a;® fl- 2y® — cr) — 2 ^ — Z)®, 
the components of the molecular rotation are first found, viz.:— 
^ V ^ 
Then the potentials L, M, N of distributions of matter of densities ~, —, x 
^ Ztt -tt Ztt 
respectively throughout the ellipsoid are determined. 
