154 Proceedings of the Royal Society of Edinburgh. [Sess. 
so that OA, OB, OC (in this order) shall form a right-handed system. We 
distinguish the positions of the corresponding lines after an interval St by 
OA', OB', OC'.* If V be the velocity of the pole along its path, and (5^ fhe 
angle between the projections on the tangent plane to the sphere of two 
consecutive tangent lines to the path, we have 
CC' = r8j^, AC'A' = 8x (1) 
At the instant t the component rotations about OA, OB, OC are 0, v, n 
respectively, and the components of angular momentum are accordingly 
0, Av, Cn . . . . . • (2) 
In the time St these are altered to 
0, A(v-\-Sv), C(n + Sn) . . . • (3) 
about OA', OB', OC', and therefore to 
- AvSx + CrivSt, A{v -f Sv), C{n + dn) . . . • (I) 
about OA, OB, OC, terms of the second order being neglected. 
If, as we will suppose, the external forces have zero moment about the 
axis of symmetry, they may be replaced by two forces P, Q acting at C 
along the tangents to the arcs CA, CB respectively, i.e. along and at right 
angles to the path of C. P is, in fact, the moment of the external forces 
about OB, and — Q that about OA. Hence, equating the increments of 
angular momentum to —QSt, VSt, 0 respectively, we find 
A«| = Q + C«., a| = P (5) 
with n = const. These are the equations which I take leave to call 
“ intrinsic,” as involving no arbitrary lines or planes of reference. 
Tlie expressions dv/dt and vd^ldt are the accelerations of the pole C 
along and at right angles to its path on the sphere. If we put 
n = 0 we have as a particular case the equations of motion of a particle on 
a spherical surface, and we infer that the motion of the pole C in the 
present case will be exactly the same as that of such a particle, of mass A, 
under the same forces P, Q, provided we introduce in addition a fictitious 
deviating force Onv acting always towards the left of the path, as viewed 
from without the sphere. j* This statement includes the old rule about 
hurrying on the precession,” but is more precise and of more general 
application, and at least equally simple. The examples which follow are 
intended mainly to illustrate its convenience. 
* The figure is simplified by the assumption that C' may be taken to lie in CA. The 
error thus involved, and in the consequent positions of A', B', is of the second order, and 
so does not affect the final results. 
t There is also an obvious interpretation of (5) in terms of the two-dimensional 
dynamics of a particle ; but this is, as a rule, less convenient for our purpose. 
