236 On Induced Stability. 
If now the plane is given a simple oscillation vertically, 
g must be replaced in the above equation by g + an 2 cos nt, 
and the motion is rendered stable if 
S j-I(t-P)M V C(0+MP) f ff (b-p)KA \ 
^12 A + M4* a J " ^A(A + M6 2 ) LC(C + M6 2 ) ^ J ' 
where, as before, in making the approximation it is assumed 
that \ , Tyi-7 2 a * s sma U- The range of action is limited by 
the conditions of the problem which require that an 2 must 
not exceed g. 
3. In general the single equation determining the oscil- 
lations of a system about steady motion is of order higher 
than the second, as for example in the case of the spinning 
top, which we now examine in the present connexion. The 
unstable position of equilibrium of a symmetrical top may be 
rendered stable either by an axial spin or by an imposed 
vertical vibration of the point of support. The two actions 
together might therefore be expected to reinforce one another, 
if either singly is not of sufficient intensity. 
For the small oscillations about the position of equilibrium 
x —py — (fj? — 2a.n 2 cos ni)x = 
y 4-jocc— (/bu 2 — 2oin 2 cos nt)y = 0, 
and a solution is 
00 
x — 2 A r sin \(c — rn)t + €\, 
OO 
00 
y = 2 A r cos {(c — rn)t + e}, 
00 
where 
- {fju 2 + (c - nif —p(c - rn) }A r + sm 2 (A r _i + A r+ i) = 0, 
This set of conditional equations is similar to the system (r) 
in § 1, and admits of corresponding approximate treatment. 
If ol is small and n is large compared with //, and p, we find 
/L6 2 + c 2 -pc-2a 2 n 2 = 0, 
and therefore for stability in this case 
2aW>/* 2 -±p 2 . 
Thus when the imposed motion is comparatively rapid the 
two actions are simply cumulative. 
Manchester, October 1907. 
