On the rolling Pendulum. 419 
= being the circumference of the circle of which the diameter 
is l. 
Again, let the same pendulum oscillate upon another cylinder, 
parallel to the first, and of the same radius with it, placed at the 
distance a’ from the centre of gravity: then, r’ being the time 
of an oscillation, we have as before 
Pe ke + c? 
a’ 
vanr/—. . 
If we suppose + = 7’, then d= 7’; and we get 
+ a —2c 
2 2 
l= : — +a— 2c 
e+e 
k= am +a’ — 2c. 
Subtract these equations aud divide by a—a’; then 
kte+c=aa’. (1) 
+e keper 
=a’, and = a: substitute 
‘ ‘ a 
From this equation, 
a’ 
these values in the expresions of /, and we obtain 
l=a+a’—2¢. (2) 
If we suppose c = 0; or, which is the same thing, if we sup- 
pose that the pendulum oscillates upon fixt axes, instead of 
rolling upon cylinders; the two foregoing equations will become 
k= 20. 
l=a+a 
Now these last equations, which are familiar to geometers, 
comprehend all the properties of the isochronous oscillations of 
a body upon different fixt axes; and, by means of the equa- 
tions (1) and (2), the like properties are extended to the case 
when the body rolls upon cylinders. It is to be observed that, 
in this investigation, it is not necessarily supposed that c, or the 
radius of the cylinder, is small: for the quantities left out in the 
general equation, were rejected on account of the smallness of 
the arc of vibration. 
We liave supposed that the cylinders are parallel ; but the 
isochronism of the oscillations does not absolutely require this 
condition. When the axis passing through the centre of gra- 
vity is parallel to a given line, 4* is determined in quantity; but 
it does not follow conversely that, when &? is given, the axis will 
have only one position. On the contrary, if we except the cases 
of a maximum or a minimum, there are many different axes (they 
will be all contained in a conic surface) that have the same 
momentum of inertia, The isochronism of the oscillations upon 
3G 2 the 
