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Proceedings of the Koyal Society of Edinburgh. [Sess. 
which the hoop rolls — one parallel to the axis of figure, one tangential 
to the hoop in the plane of rolling, and one at right angles to these 
through the centroid. These axes are represented by PC', PD', PE in 
the figure (p. 345). About PC' the angular velocity is \js + (p cos 0, and 
the angular momentum is (C + m<x 2 )(^ + 0 cos 0) ; and in consequence of 
the turning at angular velocity <j> sin 0 about PE there is a rate of 
growth (C A ma 2 )(\js (p cos 0)(p sin 0 of angular momentum about the 
instantaneous position of PD'. Again, in consequence of the turning 
with angular velocity <f> cos 6 about PC' the axis PE is moving away 
from PD', and angular momentum about PD' is growing from this cause 
at rate — A<fi 2 sin 0 cos 0. The angular velocity 0 about PD is changing 
at rate Q as PD' moves, and from this cause there is a rate of growth 
(A + ma 2 )0 about PD'. 
The total rate of growth of angular momentum about PD', or, more 
strictly, about the fixed direction with which PD' instantaneously coincides, 
is thus 
(A + ma 2 )6 + (C + mci 2 - A)</> 2 sin 6 cos 6 + (C + ma 2 )(fxj/ sin 0 ; 
and this is equal to the moment of applied forces — mga cos 0, so that 
we get the same equation of motion as before [(59) above]. In the same 
manner the <p and \[r equations might be established. 
30. As another example, we may apply this elementary method to the 
gyrostatic pendulum. Let O be the point of support, and refer to axes 
OC, that of symmetry of the arrangement of mass, OE at right angles 
to OC in the vertical plane through the axis of symmetry, and another 
OD at right angles to the plane just specified. Then find the rate of 
growth of angular momentum about OD due to the three moving axes. 
The rate of turning about OD is 6, whether we regard OD as a moving 
axis or as the fixed axis with which at the instant the moving axis 
coincides. On the other hand, the rate of alteration of 0, associated with 
the moving axis, is 0. At time dt after the instant considered, the angular 
momentum about the altered position of OD is A(6-\-0dt), and hence the 
component of angular momentum about the old direction of OD contributed 
by OD in its new position is A(6 + 6dt) cos (6dt) or A (Q + Odt). The rate 
of growth of angular momentum about the instantaneous position of OD 
at time t is thus AO. 
There are two other sources of growth of angular momentum about this 
direction. If the motion of the vertical plane containing the axis of 
symmetry, as regarded by an eye placed below the pendulum, be in the 
counter-clockwise direction, and the axes OD, OE, OC, taken in this order, 
form a usual system of axes, the axis OC is, in consequence of the motion 
