1888. ] viscous rotating cylinder. 261 
one value of a becomes zero and when, as pointed out in § 10, 
the cylinder ceases to remain secularly stable. This property 
applies to rotating liquids in general. 
16. If we proceed to the next approximation we find for the 
equation in @ 
v p\2. Il y 
a(a+2w1)+nk, —4n(n —1l)a— \y —(n—1) (=) a } = Val) 
Let us write A,, A, respectively for the two expressions 
Jo tnk, +o and jo + nk, —o. 
The values of « will be given by 
a=”, whee) 
a few 2+ nk, 
a @ =) (3 ay JIx 7 
= ih 2d, + (r= 1 or 2)...(60 
Sel er (60), 
showing that, if we retain terms of order (v/a?)?, both real and 
imaginary parts of a will be diminished and therefore the waves 
will be slightly retarded (relatively) by viscosity. 
{2r, — (n— 1) (v/a?) une 
17. We may also readily obtain first approximations to the 
values of a when v/a’ is very large. This will apply to the cases 
in which either the liquid is highly viscous or the radius of the 
cylinder very small. 
If we suppose z finite a will be very great and z will be given 
by 
{z—4n(n — 1)} (2-"2S. /z — Qe Ty fz) 
+ 4n(n—1fP2-@ey _/z =0...(61), 
the root z=0 being excluded from this approximation. The roots 
of this equation are real and positive and separated by those of 
(41). Owing to the largeness of a the relative motions will die 
away very rapidly, being quickly annulled by viscosity. 
If a be not great z will be small. Putting therefore 
= 1 
—n/2 = 
q ENE 2"T'(n +1)’ 
Zz OrVRT  /z= : 
n+1 Oe T(n fe 2) ) 
we find for «the quadratic 
a(a+2er)+nk, —2(n?—-1)av/v?=0 ......... (62). 
