222 Prof. M. J. M. Hill. [Mar. 1, 



Also p is the pressure, p the density, and V the potential of the 

 impressed forces. The minimum value of plf>+V is n//>, where n//> 

 must be determined from the initial conditions. 



Further R, are such that 



r = R sin 6, 

 z7t = Rcos0. 



The whole motion depends on the following constants : 



(1) The radius of the sphere a. 



(2) The uniform velocity Z 



(3) The minimum value ofpjp+V, viz., Tl/p. 



4. If c be not equal to a, then the surface containing the same par- 

 ticles 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 

 A; 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 pressure continuous. Hence, 

 if a motion of fluid outside the ellipsoid exist continuous with the 

 rotational motion inside, 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 Helm- 

 holtz's method determines the irrotational motion by means of the 

 distribution of vortices only when that distribution is known through- 

 out space. This is not the case of 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 ; t'. 

 starting from the values of the velocity components 



2fc 



