260 Ma Gas Bryan, On the waves on a [June 4, 
will increase indefinitely with the time, and the cylinder will not 
possess even ordinary stability. 
15. The condition for secular stability of the viscous cylinder 
is that for all integral values of n greater than unity 
The right-hand side is least if n= 2, hence we must have 
@ <mpy + OL | pO.) jee) .ceeacee eee (52). 
For a perfect liquid the condition of ordinary stability is 
2 
5 n—1 7 o fh 
A) pu ae mpy + (n Dae 
or w < 2mpy+n(n + 1) Ly sic eee (53). 
If there be no surface tension the fluid will become unstable 
for all displacements when 
It may readily be shown that when this is the case the liquid 
could not remain in the circular form unless subjected to external 
hydrostatic pressure. Without such pressure a hollow would form 
in the centre. 
If T is not = 0 the greatest angular velocity is given by 
@ = 2mpy 460 | pa. .ckc. cna eee (55) 
while in order that external pressure may not be needed 
@ < Ipy aT /p@™ o.. sa ee eee (56), 
this will happen before the cylinder ceases to possess ordinary 
stability. It will be seen from above that the greatest angular 
velocity consistent with stability is /2 times as great for perfect 
as for viscous liquid. 
If we reduce the liquid to rest and replace the centrifugal force 
by a force \’r from the axis, we find if the lquid be perfect 
— + unk, @ (elo 6 elb}e © © 6 ¢hen sv lese elu a) ele) etalsisiin (57), 
tl i \oe oe ole eer 
where k,, = 2rpy (1 — ) ++ pe NY dosiecmae (58), 
so that stability now ceases when k,=0. When the liquid is 
rotating, ordinary stability ceases when the equation in a has a 
pair of equal roots, but since the two waves travelling in opposite 
directions have not the same period this does not happen when 
