of Virtual Velocities. 275 
from uniting with the original system the series of levers and 
surfaces we have just introduced. 
Now it is easy to perceive, that the new connexion we have 
established, leaves the points at liberty to describe, backwards 
or forwards, the same indefinitely small spaces, which were 
allowed them by the original machine, provided they were 
previously at liberty to describe spaces proportional to a, a’,a’... 
in the opposite directions: but this will evidently be the case; 
for a, a, a’.... beimg indefinitely small, the equations to which 
they are subject, must be homogeneous with respect to these 
variables, or may be considered so; they will consequently re- 
main equally satisfied when these quantities simultaneously 
change their signs. 
From hence it follows, that, in the combined machine, the 
slightest tangential force applied to one alone of the points, on 
either side of it, will necessarily disturb the equilibrium. The 
equilibrium is, therefore, impossible, unless 
PPPs PP ps. i=0, 
which is the symbolic enunciation of the principle in question. 
The converse of this proposition is equally true; namely, 
that if the equation 
PB+PB+P’p’+. Ae .=0, 
be satisfied for every indefinitely small variation in the position 
of the system, there will be equilibrium. 
For if an initial motion were possible, the equilibrium 
might evidently be restored, by applying new forces in the 
opposite direction to those points only, whose variations, taken 
along the initial spaces, (but not necessarily identic with them,) 
may be regarded arbitrary and independent in other respects, 
without interfering with those points, whose variations are 
altogether determined, and their motions constrained, by those 
Vol. Il. Part Il. Nw 
