630 
GENERAL CONSIDERATIONS ON DYNAMICAL STABILITY. 
have been applied, less than the maximum one spoken of above, but considerably 
greater than that necessary to produce the initial motion, it is not difficult to see that 
this lesser pressure, by reason of its continued action on the body, may be sufficient 
to carry it through the position in which it would have required the maximum pres- 
sure to have heldxt-, for in all the positions in which this pressure exceeded that 
necessary to move the body, the excess of its work over that of the resistance is ac- 
cumulated under the form of vis viva, and in all those in which it fell short of that 
pressure, the work thus accumulated comes to its aid, and carries the body forward*. 
24. To move a body from a given position into any other, it is not necessary that 
the work of the forces whence this change of position results should continue to be 
done upon it during the whole period of such change. They may be in the nature of 
pressures whose action ceases when they have communicated vis viva necessary to 
continue the motion until the second position is attained, as in the case of the impact 
of one body on another, or of a gust of wind acting on a floating body, or of a blow 
of the sea. 
In all these cases the excess of the work of the disturbing forces over that of the 
* As an illustration of this principle, let us take the analogous case of a cylindrical body (fig. 7) — whose 
radius CD is represented by a — rolling on a horizontal plane, and to simplify the investigation let us suppose the 
force P, which causes it to roll, to be applied horizontally to the axis c. Let the weight of the cylinder be re- 
presented by W, and suppose it to be so loaded that the distance of its centre of gravity from its axis may be 
CG(=c). Let the cylinder be made to roll through an angle (S) from the position in which it would rest, and 
in which CG is vertical. The work done upon it by P, whilst it thus rolls, is represented by PaS, and the work 
opposed to its rolling by the weight of the cylinder, is WcversS, 
Pofl— Wc vers 5 
is therefore the excess of the work of the forces whose tendency is in the direction of the motion over the work 
of those whose tendency is in the opposite direction. This excess represents one-half the vis viva, and in the 
extreme position into which the cylinder rolls and from which it begins to roll back, this vanishes, so that in 
this position 
Pafl— Wc vers fl=0. 
If this position be the inverted position of the body 0=7r, and 
2Wc 
P= ■ 
Tca 
Now let us suppose that P, instead of being a constant pressure, had been made so to vary that in every 
position into which the body rolled it was only just suflficient to make it roll slowly out of that position. The 
maximum pressure P', applied under these circumstances, would have been that corresponding to the position 
in which CG is horizontal, and would have been represented by the formula 
a 
whence it folloAvs that 
P=-P'=-6366P'; 
TT 
so that the pressure, which, being continually applied the same in amount, would be sufficient to overturn the 
body, is less than two-thirds of that which must be exerted in its position of maximum effort, the lesser but 
constant pressure accumulating in its previous position, an excess of work which is sufficient to carry the body 
through and far beyond that position in which it would have required the greater pressure to have held it. 
