610 
DYNAMICAL STABILITY. 
‘‘When, being acted upon by given forces, a body or system of bodies has been 
moved from a state of rest, the difference between the aggregate work of those forces 
whose tendencies are in the directions in which their points of application have been 
moved, and that of the forces whose tendencies are in the opposite direction, is equal 
to one-half the vis viva of the system.” 
Thus, if 2^1 be taken to represent the aggregate work of the forces by which a 
body has been displaced from a position in which it was at rest, and the aggregate 
work (during this displacement) of the other forces applied to it; and if the terms 
which compose 2^^ and 2^2 be understood to be taken positively or negatively, ac- 
cording as the tendencies of the corresponding forces are in the directions in which 
their points of application have been made to move or in the opposite directions ; 
then representing the aggregate vis viva of the body hy^'Zwv- 
2^q-^-2^^2=^ 2M;y2, 
( 1 .) 
Now 2^2 I’epresenting the aggregate work of those forces which acted upon the 
body in the position from which it has been moved, may be supposed to be known ; 
2 mi may therefore be determined in terms of the vis viva, or conversely. 
2. In the extreme position into which the body is made to oscillate and from which 
it begins to return, it, for an instant, rests. In this position, therefore, its vis viva 
disappears, and we have 
2tq+ 2^2=0 . (2.) 
This equation, in which 2%^ and 2«#2 are functions of the impressed forces and of 
the inclination, determines the extreme position into which the body is made to roll 
by the action of given disturbing forces ; or, conversely, it determines the forces by 
which it may be made to roll into a given extreme position. 
3. The position in which it will finally rest is determined by the maximum value of 
2 miH- 2 m 2 in equation 1; for, by a well-known property, the vis viva of a system* 
attains a maximum value when it passes through a position of stable, and a mini- 
mum, when it passes through a position of unstable equilibrium. The extreme posi- 
tion into which the body oscillates is therefore essentially different from that in 
which it will finally rest. 
4. Different bodies, requiring different amounts of work to be done upon them to 
bring them to the same given inclination, that is (relatively to that inclination) the 
most stable, which requires the greatest amount of work to be so done upon it, or in 
respect to which 2^^ is the greatest. If, instead of all being brought to the same 
given inclination, each is brought into a position of unstable equilibrium, the corre- 
sponding value of 2 mi represents the amount of work which must be done upon it to 
overthrow it, and may be considered to measure its absolute, as the former value 
* Poisson, Mecanique, Art. 571. 
