GENERAL CONSIDERATIONS ON DYNAMICAL STABILITY. 
631 
resistances accumulates, so as to be represented at any time by half the vis viva of 
the body, and it is the work accumulated under this form and in this amount, which, 
when the operation of the disturbing- forces is withdrawn, carries the body forwards. 
25. If, in the case of an oscillatory motion, the force which causes that motion be 
intermittent, and if the periods of this intermission so coincide with those of the body’s 
oscillation that the force is withdrawn when any oscillation in its direction is com- 
pleted, and renewed when the next oscillation in that direction begins, it is evident 
that the vis viva created by it in all such successive oscillations will be accumulated, 
and the amplitudes of the oscillations rendered, in succession, greater than one an- 
other, so that by the intermittent action of a small force which so synchronizes with 
the oscillations of the body on which it is made to act, a great inclination of its posi- 
tion from that of its equilibrium may eventually result*. This might for instance be 
produced, in respect to a floating body, by the action of gusts of wind or blows of the 
waves, repeated at stated intervals. 
If the periods at which the intermittent work is done upon the body, instead of 
synchronizing with the commencement of each oscillation and thus favouring the 
motion, had so occurred as to oppose it at stated intervals (which may or may not 
synchronize with the body’s own oscillations), it is evident that a complicated motion 
dependent on these several conditions will result, in which the oscillations will be 
repeated in cycles. 
* As an illustration of this, let us suppose that in case of the cylinder (fig. 7), the motive force, instead of 
being applied horizontally to the axis, is a weight w applied to the point D on its surface when the cylinder 
was inclined at the angle fl to its position of stable equilibrium ; and that when the point D had, by the rolling 
of the cylinder, been made to descend until it again touched the plane, the weight w was withdrawn, the 
cylinder completing its oscillation on the other side of the vertical by reason of the vis viva thus communicated 
to it. 
On its return it will oscillate (the resistance being neglected) through the same angle, on the side of the 
vertical from which it started, as it completed on the opposite side. Let the weight w be then placed again 
on the point D, and taken oflf a second time when this point comes in contact with the plane ; and so continu- 
ally, a rocking motion of increasing amplitude being produced by the alternate placing and withdrawal of the 
weight until at last the position of CG is reversed and the cylinder is overturned. 
Let 02, 03 ... 9 „ be the angles on either side of the vertical through which, in its successive oscillations, the 
cylinder is made to roU, then shall we obviously have the following relations ; — 
(Wc d- tea) vers 0 =Wcvers0i, 
(Wc + ica) vers 0j=Wc vers 02, 
&c. = &c. 
(Wc + Wfl) vers 0„_i = Wc vers 0„. 
Multiplying these equations together, 
(Wc + wa)’' vers 0 = (Wc)’‘ vers 0„. 
But 0n='J!'. vers 0„=2 and vers 0 = 2 sin ^2 
(Wc-r2ca)“sin®2 
and 
