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XX. On Induced Stability. By Andrew Stephenson.* 
THE conditions under which an imposed periodic change 
in the spring of an oscillation exerts a cumulative 
effect in magnifying the motion, have already been investi- 
gated f. We shall now examine the influence of such a 
variation on instability of equilibrium and, in certain cases, 
of steady motion. 
The equation of motion about statically unstable equilibrium 
under variable spring is 
'x — (/x 2 — 2ctn 2 co$nt)x = 0, 
and the complete solution is given by 
00 
x = % A r sin {(c— rn)t + e}, 
— oo 
where 
-A r {/. 2 + (c-r7i) 2 } + «^(A r _ 1 -|-A r+1 )=0. . . (r) 
In the limit when k, taken positive, is infinite 
so that the series is convergent in both directions. If the 
eliminant of the equations (r) gives a real value for c, the 
elementary oscillations are of constant amplitude and the 
equilibrium is stable. 
An approximation t for c is readily obtained when a is 
small, for in that case, r being positive, 
A. 
H ? + {c + rnf^ ±{r ' 1) 
approximately, and on substituting for A ± and A_x in (0) we 
find that for a real c when a is small n must be large compared 
with fi : then 
c 2 = 2a 2 /i 2 -y, 
and for stability n must be greater than fjbjoL^/2. 
Thus the system can always be maintained about a position 
of otherwise unstable equilibrium by a periodic variation in 
spring of sufficiently large frequency. The inverted pen- 
dulum, for example, is rendered stable by rapid vertical 
* Communicated by the Author. 
t " On a class of forced oscillations/' Quarterly Journal of Mathe- 
matics, no. 168, 1906. " On the forcing of oscillations by disturbances 
of different frequencies," Phil. Mag. July 1907. 
% Similar to that employed in the former of the two papers referred 
to above. 
