1888. } viscous rotating cylinder. 257 
hereafter that the former is the case if the liquid be but slightly 
viscous, the latter if the viscosity is exceedingly great. 
10. It is physically impossible that the equation (33) should 
be satisfied by a purely imaginary value of z, and therefore of a 
except when v=0. In fact a strictly periodic motion can never 
exist in the case of viscous liquid unless energy be supplied from 
without. For after a complete period the liquid would return to 
its original state, and its energy would therefore be the same as at 
the beginning. But all motions of viscous liquids except rigid 
body displacements are accompanied by dissipation of energy 
which is converted into heat. Such energy, since it is not derived 
from the system, would have to be supplied from external sources 
as, for example, in the case of periodic forced waves. 
This does not prevent the possibility that the real part of z or 
a may be negative, provided that the total energy for given angular 
momentum be not a minimum in the state of relative equilibrium. 
In this case the dissipated energy will be derived from the system, 
which will pass into configurations in which the total kinetic and 
potential energy is less than in the original state. 
From this it is evident that in any case of rotating viscous 
liquid, if we refer the motion to axes rotating with the fluid mass 
and suppose the small displacements of the liquid particles to vary 
as e “, the real part of « can only change from positive to negative 
when @ vanishes. The waves then become stationary corrugations 
on the surface, and for such displacements the total energy ceases 
to be a minimum. Accordingly if one of the waves becomes 
unstable, this must happen when the energy criterion for secular 
stability ceases to be satisfied. 
In the case of the circular cylinder one value of @ will vanish 
and change sign only when &, =0, and it may readily be verified 
that for minimum energy /,, must be positive. How the conditions 
of stability are altered by a complete absence of all viscosity will 
be best seen by considering the case in which the viscosity of the 
liquid is supposed relatively small. 
11. Let us, then, suppose either that the kinematical viscosity 
y is very small or else that the radius a@ of the cylinder is very 
great so that v/a? is small in comparison with or /yp. Then 
the roots of (33) may be expanded in powers of v/a’. We must 
either suppose z to be finite and a small, or @ finite and z very 
large. 
12. If we adopt the first hypothesis we find that the values 
of z are ultimately given by the equation 
ees. Jz = QM lz ——i (Pane ae eae (41), 
