1887.] Equations to the Motion of a number of Cylinders, déc. 139 
A Circular Cylinder. 
3. Let a be the radius, and o the density of the cylinder, then 
2 =4na (p+c) (#’+y’), 
y= a log {(r’ cos 0 — x)? + (7’ sin 0 — a), 
K , K ’ : q 
= oe oe Ni guea (x cos 0 + ysin @') + &e. 
Whence A=—«u/27, W=-—xKy/2r. 
Taking for a moment the origin at the centre of the moving 
cylinder, the value of ‘N is 
N=- [z dr d0 = 5 | [rdrd = }u(c*— a’), 
whence ‘N is constant, therefore disappears on differentiation. 
The value of ZL may consequently be written 
L = pra'(p +o) (a + yf) + 2ep (dy — ¥2); 
and the equations of motion are 
wa (p+o)é+Kpy =X (10) 
ma (p+a)y —Kp4= Y 
which agree with the equations obtained by Prof. Greenhill*. If 
no impressed forces act, the cylinder, as was first shown by Lord 
Rayleigh+, will describe a circle in the same direction as that of 
the cyclic motion. 
An Elliptic Cylinder. 
4, If x,— a +4(y,—y') =acos (E —«n), 
where #,, y, are current co-ordinates referred to Oz’, Oy’, then K&/27 
and — «n/27 are the velocity potential and current function for 
cyclic motion round an elliptic cylinder. 
Let N= (v,+ vy,)/a, w= (xe +12y')/a, 
then since cos #=—+ log (w + /a*—1), 
we have £—in=— clog {A—p+./(A—p)*— 1} 
=— vlog 2X —+4 log (1- et oy) 
=—1 (log 2r, + 8,) 7 (a + wy’) (cos 8, — vsin @,). 
* Mess. Math. Vol. 1x. p. 115. + Ibid. Vol. vir. p. 14. 
