TRANSACTIONS OF SECTION A, 439 
The following Papers were then read :— 
1. Helium and Radio-activity in Common Ores and Minerals, 
By Hon. R. J. Strutt, 7B. 
2. On the Motions of Ether produced by Collision of Atoms or Molecules 
containing or not containing Electrons.| By Lord Ketvin, G.C.V.O., 
F.R.S. 
3. On Secular Stability. By Professor Horace Lams, /.B.S. 
The author showed an experiment intended to illustrate the distinction 
between ‘ordinary’ or ‘temporary’ stability and ‘permanent’ or ‘secular’ 
stability, to which attention has been drawn by Poincaré in his research on 
figures of equilibrium of rotating liquid. A pendulum, symmetrical about its 
axis, hangs by a Hooke’s joint from the lower end of a vertical spindle, which can 
be made to rotate by means of a pulley with constant angular velocity #. When 
o =0, the vertical position is of course stable, and the two normal modes of small 
oscillation have equal periods. For the present purpose these modes may be 
analysed into two circular vibrations in opposite directions. When the spindle is 
made to rotate, the usual method of ‘small oscillations,’ which ignores the effect of 
dissipative forces, leads to the conclusion that the vertical position is still 
‘ ordinarily ’ stable for all speeds, the only effect of the rotation being that the two 
circular vibrations have now different periods, that being the more rapid whose 
direction of revolution agrees with that of the shaft. The criterion of ‘ secular’ 
stability imposes, however, a limit to the valueof . If A, A, C be the principal 
moments of inertiaof the pendulum at the joint, M the mass, / the depth of the 
centre of gravity below the joint, the condition of secular stability is 
2 Mgh 
o< yee BE 
If w? exceed this limit, a new position of relative equilibrium is possible, in 
which the pendulum would rotate at a constant inclination, 6, to the vertical, 
given by 
Mgh : 
(A —C)o*’ 
this position is ‘ secularly’ stable, and the former now unstable. 
The latter conclusion is confirmed and explained if we work out the question 
by the usual method of small oscillations, introducing, however, frictional forces 
proportional to the relative velocities at the Hooke’s joint. It then appears that, 
as regards the two circular vibrations, the amplitude of the one whose direction 
agrees with that of the shaft exponentially increases (when w? exceeds its critical 
value), whilst that of the other diminishes. Hence the pendulum if disturbed 
ever so little from the vertical position describes an ever-widening conical path 
in the direction of the rotation of the shaft, tending towards the second position 
of relative equilibrium above referred to. 
cos 6= 
FRIDAY, AUGUST 2. 
Discussion on the Constitution of the Atom, opened by 
Professor E. RutrHerrorp, £2.58. 
1 Electrician, August 16, 1907; Phil. Mag., September 1907, 
