1895-96.] Lord Kelvin on the Motion of a Liquid. 
119 
On the Motion of a Heterogeneous Liquid, commencing 
from Rest with a given Motion of its Boundary. 
By Lord Kelvin, President. 
(Read April 6, 1896.) 
I use the word “liquid” for brevity to denote an incompressible 
fluid, viscid or inviscid, but inviscid unless the contrary is ex- 
pressly stated. A finite portion of liquid, viscid or inviscid, being 
given at rest, within a bounding vessel of any shape, whether 
simply or multiply continuous ; let any motion be suddenly pro- 
duced in some part of the boundary, or throughout the boundary, 
subject only to the enforced condition of unchanging volume. 
Every particle of the liquid will instantaneously commence moving 
with the determinate velocity and in the determinate direction, 
such that the kinetic energy of the whole is less than that of any 
other motion which the liquid could have with the given motion 
of its boundary.* This proposition is true also for an incompres- 
sible elastic solid, manifestly ; (and for the ideal “ ether ” of Proc. 
R.S.P., March 7, 1890 ; and Art. xcix. vol. iii. of my Collected 
Mathematical and Physical Papers). The truth of the proposition 
for the case of a viscous liquid is very important in practical 
hydraulics. As an example of its application to inviscid and 
viscous fluid and to elastic solid consider an elastic jelly standing 
in an open rigid mould, and equal bulks of water and of an inviscid 
liquid in two vessels equal and similar to it. Give equal sudden 
motions to the three containing vessels : the instantaneous motions 
of the three contained substances will be the same. Take, as a 
* Cambridge and Dublin Mathematical Journal , Feb. 1849. This is only a 
particular case of a general kinetic theorem for any material system whatever, 
communicated to the Royal Society, Edinburgh, April 6, 1863, without proof 
( Proceedings , 1862-63, p. 114), and proved in Thomson and Tait’s Natural 
Philosophy , sec. 317, with several examples. Mutual forces between the con- 
taining vessel and the liquid or elastic solid, such as are called into play by 
viscosity, elasticity, hesivity (or resistance to sliding between solid and solid), 
cannot modify the conclusion, and do not enter into the equations used in 
the demonstration. 
