Virtual Velocities.
51. This principle, discovered by John Bernouilli, and called the principle of virtual velocities, is perfectly general, and may be expressed thus:—
If a particle of matter be arbitrarily moved from its position through an indefinitely small space, so that it always remains on the curve or surface, which it ought to follow, if not entirely free, the sum of the forces which urge it, each multiplied by the element of its direction, will be zero in the case of equilibrium.
On this general law of equilibrium, the whole theory of statics depends.
52. An idea of what virtual velocity is, may be formed by supposing that a particle of matter m is urged in the direction mA by a force applied to m. If m be arbitrarily moved to any place n indefinitely near to m, then mn will be the virtual velocity of m.
53. Let na be drawn at right angles to mA, then ma is the virtual velocity of m resolved in the direction of the force mA: it is also the projection of mn on mA; for
mn:ma::1:cos nma and ma=mn cos nma.
54. Again, imagine a polygon ABCDM of any number of sides, either in the same plane or not, and suppose the sides MA, AB, &c., to represent, both in magnitude and direction, any forces applied to a particle at M. Let these forces be resolved in the direction of the axis ox, so that ma, ab, bc, &c. may be the projections of the sides of the polygon, or the cosines of the angles made by the sides of the polygon with ox to the several radii MA, AB, &c., then will the segments ma, ab, bc, &c. of the axis represent the resolved portions of the forces estimated in that single direction, and calling α, β, γ, &c. the angles above mentioned,
ma=MA cos α; ab=AB cos β; and bc=BC cos γ,
