234 Mr. A. Stephenson on 
vibration of the point of support. The above analysis shows 
that quite apart from gravity, an imposed motion of small 
amplitude and high frequency produces a comparatively 
slow simple oscillation about its own direction ; the impressed 
action must therefore exert a restoring moment on the body 
proportional to the displacement of the mass centre from the 
line of the applied motion, and if this moment numerically 
exceeds the outward moment due to gravity stability is 
•ensured *. 
If the pivot is given a simple vibration of amplitude a with 
frequency n per 2ir seconds, the equation of motion about 
the unstable position is 
For a simple pendulum 1 metre long fi 2 = 10 approximately, 
and if a =10 cm. for stability n>44*7; thus a frequency of 
7*2 per second is sufficient to maintain relative equilibrium. 
If the pendulum is 20 cm. in length /m 2 = 50, and if a = 10 
the condition would give n>20, a frequency of over 3*2 
per second. In this case, however, fi/n is not sufficiently small 
for the approximate formula to be applicable. 
There is no difficulty in verifying the stability under the 
imposed motion by experiment. It is found furthermore that 
the pendulum may be rendered approximately steady in a 
position sensibly oblique by a comparatively small inclination 
of the direction of vibration, and it is of interest to enquire 
under what circumstances this occurs. 
If the path of the pivot makes a small angle fi w T ith the 
vertical the equation of motion is 
* ~~ L2 {.9~ an2 cos n x = fi~r 2 an2 cos nt - 
For the particular solution giving the forced motion due 
to the disturbance represented by the term on the right, we 
have 
00 
x = XB r cos rnt, 
o 
v^here 
-B r g+rv)+igV(B,_ 1 + B, +1 )=0, . . (,•)' 
* The stability of this particular system is worked out from first 
^principles for a special case in a paper " On a new type of dynamical 
stability," read before the Manchester Literary and Philosophical Society 
in January 1908. 
