372 
Proceedings of the Royal Society 
pounded with the given rotational motion. The irrotational motion 
in the case of the spherical hollow is of course easily calculated by 
the well-known spherical harmonic analysis for fluid motion. We 
consider here only the instantaneous motion, which exists at the 
instant when the impulse is completed. The infinitely more 
difficult problem of working out the consequences according to any 
prescribed conditions as to force, or as to changing shape, for the 
boundary, we do not follow at present. It will be fully followed up 
for the case in which the boundary of the liquid is spherical or 
ellipsoidal to begin with, and is constrained to be always exactly 
ellipsoidal. It will be proved that in this case the molecular 
rotation of the fluid remains always homogeneous. We shall see in 
fact that the geometrical “ strain ” is essentially homogeneous 
throughout a liquid contained within a changing ellipsoidal boundary, 
provided that the motion of the fluid be either wholly irrotational, 
or be at any one instant homogeneously irrotational. The homo- 
geneousness of the geometrical strain being established, it follows 
from Helmholtz’s fundamental principles of vortex motion, that the 
molecular rotation must contiuue homogeneous ; its magnitude, when 
there is any stretching or contraction in the axial direction, varying 
inversely as the length of a line of the substance in this direction, 
and the axial direction varying so as to keep always along the same 
substantial line. 
If there is the slightest deviation from exactness in the ellipsoidal 
figure, the homogeneousness of the rotation of the liquid is not 
maintained, and there is no limit to the amount of deviation from 
homogeneousness which may supervene in consequence of motions 
which may be given to the boundary, whether in the way of change 
of shape, or of motion without change of shape. Confining our 
attention for the present to motion of the boundary without change 
of shape, we find it interesting to remark that we may go on 
indefinitely increasing or indefinitely diminishing the energy of the 
fluid motion by properly arranged action in the way of moving the 
containing vessel. To continually increase the energy I believe the 
following rule may be correct, although I do not yet see a perfect 
proof of it. Suppose the containing vessel to be given at rest, and 
the liquid within it to have perfectly homogeneous rotation within 
the not exactly ellipsoidal hollow, watch it for a little time — it may 
