On the rolling Pendulum. Al] 
when in action, lowers the surface of the water upon which it 
floats, about one inch in an hour; it is stopped merely by blow- 
ing air into the syphons, or by preventing the efflux of water in 
any other manner. 
London, Noy. 1821. C. A. Buspy. 
P.S. I made the first, and comparatively imperfect, model of 
my hydraulic orrery at New-York, where it was seen in action 
by the Mathematical Professor Dr. Adrian, of Columbia College, 
Dr. David Hosac, F.R.S., Dr. Samuel L. Mitchill, Dr. Mac 
Neven, and many other leading members of the American Phi- 
losophical Society, estaolished by the Legislature of the State in 
that beautiful and flourishing city. C.A.B. 
LXXXIV. On the rolling Pendulum. By Jamus Ivory, M.A. 
F.R.S. : 
1E is proposed to demonstrate that the properties discovered by 
Huyghens, concerning the isochronous vibrations of a solid body 
about different fixt axes, are likewise true when the body rolls 
upon cylinders, provided the cylinders roll without sliding. 
Let ¢ denote the radius of the cylinder upon which the pen- 
dulum rolls; a, the distance of the axis of the cylinder from the 
centre of gravity of the pendulum; ¢ the angle which a plane 
passing through the axis of the cylinder and the centre of gravity 
makes, at the time ¢, with a vertical plane likewise passing 
through the same axis. The whole mass of the body being m, 
let dm denote a molecule; x the distance of dm from the hori- 
zontal plane in which the axis of the cylinder moves; and y its 
distance from a fixt vertical plane, suppose that containing the 
axis of the cylinder and the centre of gravity wheu ¢ = 0, or 
when the pendulum is at rest. For the sake of simplicity, we 
shall suppose that the whole matter of the pendulum is concen- 
trated in a straight line; this supposition being made merely to 
abridge algebraic expressions; for nothing is easier than to ex- 
tend what is proved in this case to a body of any figure. 
Now, the accelerations impressed upon dm by its connection 
ddxz ddy , 
a Pala and the acceleration re- 
ceived from gravity is g; g denoting the velocity acquired by a 
falling body in one second: therefore, if 6% and éy denote vari- 
ations subject to the law of the motion of the molecule, we shall 
have this equation from the known principles of dynamics, viz. 
Sdm.}g— dl? Ld —Sdm. amt by = 0; 
with the pendulum are 
dt § “dt? 
the symbol S denoting an integration to be extended to all the 
Vol, 58, No, 284, Dec. 1821. 3G molecules 
