Ball — -On the Oscillations of a Rigid Bodij. 
11 
4. It will economize the oblique rays of light ; 
5. It will be purely achromatic ; 
6. It will light up the interior of a partially transparent object ; 
7. It will improve definition ; 
8. It is easy of application ; 
9. It will not be an expensive addition to the microscope. 
ly. Olf THE SMALL OSCILLATIONS OP A ElGID BoDY ABOUT A FiXED PoiNT 
FN"DEii THE Action op ant Forces, and, moee particulaely, when 
G-EAVITY IS the only FoECE ACTING. By EoBEET StAWELL BaLL, 
A. M., M. E. I. A., Professor of Applied Mathematics and Me- 
chanism, Eoyal College of Science for Ireland. [Abstract.] 
[Read January 24, 1870.] 
A EiGiD body rotating about a fixed point may be moved from one 
position to any other position, by rotation around an axis termed the 
' axis of displacement' through an angle termed the ' angle of displace- 
ment.' This is a well-known theorem. 
The equations of motion, being linear, depend, as usual, upon a cubic 
equation. The roots of this cubic give criteria as to the nature of the 
equilibrium. 
A 'normal axis' is defined to be '*a direction passing through the 
fixed point, about which the body will oscillate as about a fixed axis, 
when the initial ' axis of displacement' and instantaneous axis coincide 
with this direction." 
When the roots of the cubic are all real, positive, and unequal, 
there are three 'normal axes,' and small oscillations of the body are 
compounded of vibrations around these three axes. Hence we infer the 
general theorem. 
' If a rigid body, rotating around a fixed point, perform small 
oscillations about a position of stable equilibrium under the action of 
any forces, its motion is produced by the composition of vibrations 
around three fixed axes passing through the point, and each of the 
vibrations about the fixed axes is performed according to the same law 
as the vibration of the common pendulum.' 
If only one of the roots of the cubic be a real positive quantity, 
and if the initial ' axis of displacement' and instantaneous axis coincide 
with the ' normal axis' corresponding to this root, equilibrium is stable 
relative to such a displacement, but for any other initial ' axis of dis- 
placement' or instantaneous axis the equilibrium is unstable. 
If two of the roots of the cubic be real, positive, unequal quantities, 
while the third is negative, and the ' normal axes' corresponding to the 
two positive roots be constructed, then if the initial ' axis of displace- 
ment' and instantaneous axis lie in the plane containing the normal 
axes, the equilibrium is stable, while if either of these axes be not con- 
tained within this plane the equilibrium is unstable. 
