1887.] Equations to the Motion of a number of Cylinders, dc. 143 
CASE (i). Since p is positive the roots will always be real if 
M'> M 
and Kp <me (M — M)g. 
In this case the liquid is denser than the cylinder, and one of 
the roots of (16) will be positive and the other negative, and the 
positive root will be numerically greater than the negative root. 
Hence there will be two cases of steady motion, in one of which 
velocity of the cylinder will be in the same direction as that of 
the liquid, due to the circulation at points between the cylinder 
and plane ; and in the other the velocity will be in the opposite 
direction ; also the velocity in the former case will be greater than 
in the latter. 
CASE (ii). M>M, «’p>4mc (M — M) g. 
In this case the roots of (16) will be both real and positive 
provided (17) is satisfied; hence the velocity in the two cases of 
steady motion will be in the same direction as that due to the cir- 
culation. 
CASE (iil). M>M. 
In this case the cylinder is denser than the liquid, and the 
roots of (16), if real, must be both positive, hence the two 
velocities must be in the same direction as that due to the cir- 
culation. 
CAsE (iv). If either g=0 or M=WM’, (17) becomes 
apc coth’ a> p. 
Here both roots of (16) are positive, and the two velocities 
must be in the same direction as that due to the circulation. 
This case has been discussed by Mr W. M. Hicks*. 
CASE (v). Suppose that the cylinder is reduced to rest, and then 
let go. Since w and v are initially zero, the initial acceleration is 
1 ; 2 
0=— Fp, ttre (M— M’) 9 + x’p} aicheleveieieveye (18). 
Hence if the liquid is denser than the cylinder it is possible 
for the right-hand side to vanish ; in which case the cylinder will 
remain in equilibrium under the combined action of gravity 
and the pressure due to the cyclic motion. 
If the plane formed the upper boundary of the liquid the sign 
of g in these five cases would have to be reversed. 
* Quart. Journ, Vol. xvi. p. 194. 
