&c. and the sum of these partial forces will be
MA cos α+AB cos β+BC cos γ+&c.=0
by the general property of polygons, as will also be evident if we consider that dm, ma, ab lying towards o are to be taken positively, and bc, cd lying towards x negatively; and the latter making up the same whole bd as the former, their sums most be zero. This it is evident, that if any number of forces urge a particle of matter, the sum of these forces when estimated in any given direction, most be zero when the particle is in equilibrio; and vice versâ, when this condition holds, the equilibrium will take place. Hence, we see that a point will rest, if urged by forces represented by the sides of a polygon, taken in order.
In this case also, the sum of the virtual velocities is zero; for, if M be removed from its place through an infinitely small space in any direction, since the position of ox is arbitrary, it may represent that direction, and ma, ab, bc, cd, dm, will therefore represent the virtual velocities of M in directions of the several forces, whose sum, as above shown, is zero.
55. The principle of virtual velocities is the same, whether we consider a material particle, a body, or a system of bodies.
Variations.
56. The symbol δ is appropriated to the calculus of variations, whose general object is to subject to analytical investigation the changes which quantities undergo when the relations which connect them are altered, and when the functions which are the objects of discussion undergo a change of form, and pass into other functions by the gradual variation of some of their elements, which had previously been regarded as constant. In this point of view, variations are only differentials on another hypothesis of constancy and variability, and are therefore subject to all the laws of the differential calculus.
57. The variation of a function may be illustrated by problems of maxima and minima, of which there are two kinds, one not subject to the law of variations, and another that is. In the former case, the quantity whose maximum or minimum is required
