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Proceedings of the Royal Society of Edinburgh. [Sess. 
so that the integrability conditions are fulfilled. Thus it is possible to 
proceed in the ordinary way by calculating dT/dr and subtracting it from 
mr. The function of r involved in (44) and (45) is the same, and so in the 
latter case the ordinary process remains applicable, though then, appar- 
ently, the integrability conditions seem unfulfilled. This explains why in 
many cases, e.g. in the next example, when specialised axes are taken, the 
ordinary method is applicable, while the other, set forth in § 9 above, is 
not. The latter can only be applied when the values of u, v, . . . . are 
perfectly general. 
20. As a first example we take the gyrostatic pendulum problem 
referred to above. The pendulum as ordinarily made is a rigid body 
symmetrical about a longitudinal axis, and containing a fiy- wheel with its 
axis of rotation along the axis of symmetry. The suspension is by a 
Hookes joint, or by means of a piece of steel wire so short that it may be 
taken as untwistable, while yielding equally freely to bending forces in all 
vertical planes containing the wire. 
Let 6 be the inclination of the axis of symmetry to the vertical, (p the 
angle which the vertical plane through this axis makes with a fixed plane 
through the vertical containing the point of support, \js the angle which a 
plane containing the axis of the fly-wheel, and fixed in the wheel, makes 
with an axial plane fixed in the pendulum. We shall not suppose in the 
first instance that the pendulum, apart from the fly-wheel, is symmetrical, 
but take C as its moment of inertia about the axis of the wheel, which we 
shall suppose to be a principal axis of moment of inertia, and A and B as 
the other two principal axes for the point of support. We shall also denote 
the moment of inertia of the fly-wheel about its axis by C', and its moment 
of inertia about any axis at right angles to this through the point of sup- 
port by A'. It is easy to show that the angular velocity of the pendulum 
(apart from the fly-wheel) about the axis of symmetry is — <^>(l-cos0). 
That of the fly-wheel about the same axis is \js — <p(l — cos 0). 
21. We suppose now that the principal axis about which the moment 
of inertia is A is inclined at the instant under consideration at an angle <p 
to the vertical plane containing the axis of the fly-wheel, so that if without 
other change of position of that plane the axis of rotation of the wheel 
were brought to the vertical this principal axis would lie in the plane 
from which <p is measured. The pendulum is turning with angular velocity 
0 about an axis perpendicular to the vertical plane through the axis of 
the fly-wheel, and with angular velocity <p sin 0 about an axis in that 
plane and at right angles to the fly-wheel axis. The angular velocities 
about the axes of A and B (taken as related to the third axis as the 
