258 Mr G. H. Bryan, On the waves on a [June 4, 
which has already been investigated. Let z, be any root of this 
equation and let us write z,+ for z in (83), where wz is small. 
Then we find approximately 
5a (z, Sete Ties WES Ty, 22, +22 J vss nl Z) Se 
= 4n (n ca 1)*2,2)— pais ae lz, 
Wee Meow ae eae ....(43). 
Now 4, is a root of (41), also we have the well-known relation* 
Di (aiits i) Sie) 2s) iene) ean ere en (44), 
whence 
7 ie pe ee vz, aa 22,7 Ge) i Jee WE 
=(1- =) a4 PET. dy vueees (45). 
0 
Therefore neglecting the fourth power of v/a’ 
_ 8n(n—1)%2," nk,v*/a' — 2012,r*/a° 
—  (4,-4n) Wh? + 4072,70"/a" 
Now the roots of (41) are all greater than 4m. Hence the imagi- 
nary part of a which is a quantity of order (v/a*)’ is essentially 
negative. From the results of (ai) § 9, it follows that the real 
part of « must still be negative even where the viscosity is no 
longer small. Hence all but two of the wave motions travel 
round in the negative direction relative to our rotating axes, 
i.e. rotate more slowly than the liquid, but if v/a? be even mode- 
rately small the disturbance will have died away long before its 
position relative to the liquid mass has changed through an 
appreciable angle. 
Again, since h’a’J, (ha) — 2haJ,,,(ha) is of order (v/a*)*, there- 
fore by (28) Aa’ + BJ, (ha) is also of order (v/a’)’. 
Thus the height of the corrugations of the surface is very 
small compared with the components of relative displacement of 
the fluid particles in the interior. 
Unless therefore the liquid possess considerable viscosity or 
the cylinder be not very great in diameter, the slow motions cor- 
responding to these roots are not waves at all but small vortex 
motions in the interior of the liquid which die away very slowly, 
and can never be unstable, nor increase with the time. 
13. In the true waves a remains finite and therefore z becomes 
Be: 
very large when is small. We must therefore use Lommel’s 
* Lommel, p. 8. 
