12 
Proceedings of the Boyal Iriah Academy. 
If the forces wliicli act on the body have a potential, a simple geo- 
metrical construction determines the ' normal axes.' 
The quantity of energy necessary to give the body a certain dis- 
placement can be expressed in terms of the 'angle of displacement/ 
and the direction cosines of the * axis of displacement.' If along 
every radius passing tlirough the fixed point a length be measured 
from the fixed point proportional to the displacement which a given 
quantity of energy could produce about that axis, the locus of the 
extremities of these radii vectores is called the * ellipsoid of equal 
energy.' 
The greatest and least axes of the ' ellipsoid of equal energy' are the 
directions about which the same quantity of energy would produce the 
greatest and least eff'ects. 
For a displacement from the position of equilibrium around any 
radius vector of this ellipsoid, the moment of the forces acts in the plane 
conjugate to that radius vector. 
The * normal axes' are the three common conjugate diameters of 
the momental ellipsoid, and the ' ellipsoid of equal energy.' 
The length of the simple pendulum isochronous with the vibration 
about each axis is proportional to the square of the ratio of the corre- 
sponding diameter in the ' ellipsoid of equal energy' to that of the mo- 
mental ellipsoid. 
When the times of vibration about two of the ^ normal axes' are 
identical, the construction becomes indeterminate for these axes, and 
every direction in the plane conjugate to the third normal axis is a 
normal axis. 
When the times of vibration about three axes not in the same plane 
are equal, every direction passing through the fixed point is a ' normal 
axis,' and the motion of the body is isochronous, whatever be the 
initial circumstances. A body thus related to the forces which act on 
it may be called an 'universal pendulum.' 
In the ' universal pendulum,' the plane which contains the initial 
instantaneous axis and ' axis of displacement' continues to contain 
them throughout the motion. 
Every point of the 'universal pendulum' describes an infinitely 
small ellipse. 
The small oscillations of a rigid body suspended from a fixed point 
different from its centre of gravity, and acted upon by gravity, but by 
no other force, is discussed : this is the question of the conical pendulum 
in its most general form. 
The fixed point being joined to the centre of gravity, a plane is to 
be drawn in the momental ellipsoid conjugate to this line. This 
plane is called the ' conjugate plane.' 
The body may be displaced from a position of equilibrium to any 
given adjacent position by rotation around a certain axis through a 
small angle, this axis lying in the ' conjugate plane.' 
For small oscillations it is necessary that the instantaneous axi& 
should always lie in the ' conjugate plane.' 
