1888. ] viscous rotating cylinder. 255 
The radial velocity of the fluid in these undulations is given by 
ay 
00° 
displacement, are, at any time, everywhere proportional to 
{(@"/a" + RY + Ro} /r, 
Thus its amplitude, and therefore also that of the radial 
ie, to * mod {4r" + BJ, (hr) 
When 7 is small this becomes approximately proportional to 
r". Now the slow motions corresponding to n=O or 1 are not 
waves at all. Excluding these we see that the displacements of 
the fluid particles diminish indefinitely as r approaches zero, and, 
the greater be the number of corrugations (n), the more rapidly do 
they become insensible towards the centre. 
9. We now proceed to discuss a few results which hold with 
regard to the fundamental equation for z 
{2(z+2ea71/v)+-nk,a'/v?—4n(n—1)2} (27"? J, B= Digg or EDS ele) 
SEL GND aed PLE RIE =e ena (33). 
It will be necessary to remember* that the roots of the 
equation CTE 0 ee eA (39) 
are all real and positive, and are separated by those of 
Zs aa iin) hi) ae bays 0. Meas: op (40). 
(i) The equation (33) has an infinite number of roots, but 
since, in it, 2””J, /z occurs multiplied by a quadratic function 
of z, it has two and only two more roots than has the equation (39). 
(ii) The equation (38) cannot have a real root other than 
zero unless w = 0. 
For if we suppose z real and » not=0 and equate the real 
and imaginary parts of (83) to zero we find that unless z=0 we 
must have “eee fo 7a aa) 4 Vib Fy iz | AN Neo ts (41), 
and also ip CIES fog \ = a eae ee (40), 
whence also Zr AR ENA TS EE A Bh (39), 
which is impossible since the equations (39), (40) cannot have a 
common root. 
* Lommel, Studien iiber die Bessel’schen Functionen, p. 6%. 
