1888.] viscous rotating cylinder. 259 
expressions for the development of J, Jz J Ged Jz in negative powers 
of ./z*, from which it appears that ultimately 
J, Jz= (2) cos (ve -~F-n5) 
ae — ee sin (Jz-7 fe eile n3) 
Hence, if we neglect powers of v/a” higher than the first, equation 
(31) gives the following quadratic for a, 
a (4+ 200s) + mk, — “Yn (n— 1) a=0 ia (48). 
Solving for a, we find to the first order 
esa 22 0) —1) _ 
Sees 6 oS = ; ere ae (49), 
the upper or lower sign being taken throughout. 
14. If &, is positive the real parts of both values of « given 
by (49) are positive and therefore the waves diminish indefinitely 
as the time increases. For wave motions of the type considered 
the cylinder is secularly stable. 
If &, is negative the real part of one of the values of a becomes 
negative and the corresponding wave continually increases until it 
is no longer small, therefore the cylinder is secularly unstable. 
Since the real part of a is proportional to ~,, it follows that 
a 
the smaller we suppose this quantity the more slowly will the 
waves increase or diminish. 
If the liquid be perfect and w* + nk, be positive, the values of 
a will be purely imaginary. Hence the waves will be strictly 
periodic and will neither increase nor diminish. The cylindrical 
form will, for such wave displacements, be stable, but will possess 
only what M. Poincaré calls “ordimary” stability. There will be 
no tendency to return to or depart from the undisturbed state of 
steady rotation and any waves produced by the disturbance will 
continue permanently. 
If however w’+ nk, be negative, the values of a will be complex 
even ifvy=0. We now obtain 
Eo as |) SP BIS) omonoscunnesn: (50). 
The real part of one root is positive but that of the other is 
negative. Hence even if the fluid be inviscid one of the waves 
* Studien, § 17, p. 57, 
