345 
1908-9.] On Lagrange’s Equations of Motion. 
28. The last equation is written down for the sake of the comparison of 
the Lagrangian method, modified or not, with the following elementary, 
and I think instructive, method of solving gyrostatic and other rotational 
problems. It depends on the single principle, which can be demonstrated 
in a moment, that if a directed quantity L be associated with an axis, OA 
say, which is turning towards a second axis, OB say, at right angles to the 
first, with angular velocity w, there is, as a consequence, a rate of growth 
of the quantity in the direction OB of amount Lw. If there is already 
associated with this second direction, at the instant considered, a component 
of the same quantity M, which is changing at rate M, the total rate of 
growth of the vector associated with the instantaneous direction of OB is 
M + Lco. [It is to be understood that M is taken following the direction 
OB as that revolves keeping itself at right angles to OA.] This, of course, is 
a particular case of the proposition that if L x = L x cos Q 1 + L 2 cos 0 2 + L 3 cos d 3 , 
then 1b x — Lj cos 0 1 + L 2 cos 0 2 + L 3 cos 0 3 — L sin dj . 6 X — . . . . But it forms 
by itself a simple rule, immediately evident, and applicable to all kinds of 
rotational problems. 
The simplest possible example of this is a particle of mass m moving in 
a circle of radius r with speed v. At a point P on the circle the particle 
has momentum mv in the forward direction along the tangent. But the 
tangential direction of motion of the particle is turning towards the radius 
at P with angular velocity v/r. Hence the rate of growth of momentum 
in the direction from P to the centre is mvv/r = mv 2 /r. 
If, as there may be, there are two different directions OA, perpendicular 
to one another and to OB, with which are associated directed quantities of 
the same kind L, L', and these are turning towards OB with angular 
velocities <*/, there will exist in consequence a rate of growth of the 
quantity along OB of amount Lco + LV in addition to M. 
29. Now apply this principle to the problem of the hoop. Take 
axes through the point of contact P with the horizontal plane on 
