It will, however, sometimes be possible to disturb one part of the system without affecting other parts; or the system may consist of several parts, each one of which it may be possible to disturb in such a manner as not to affect the other parts. In this case it is manifest, that the equation of virtual velocities will furnish us as many equations between the forces, as there are independent parts of the system. Now two points are independent when no geometrical relation exists between their virtual velocities. Wherefore, in using the equation Σ(Fds) = 0, we must find, from the geometrical properties of the system, as many of the quantities S8, S8, S8,... in terms of the others as possible, and substitute them in the equation; the virtual velocities which are still left in it are independent, because no geometrical relation exists among them; and, therefore, the corresponding parts of the system admit of independent disturbance; we must consequently equate the coefficients of each of these terms to zero. The resulting equations are the conditions of equilibrium.

To illustrate what is here meant, we will solve the two following problems by the principle of virtual velocities.

115. A particle rests upon a plane curve line, being acted on by two forces X, Y parallel to the co-ordinate axes: to find the conditions of equilibrium.

Let y = f(x) be the equation of the curve, a, y being the co-ordinates of the position of equilibrium of the particle. Then since after the disturbance the particle still remains upon the curve, if y+dy, and a + dx be the co-ordinates of its new position they must satisfy the equation of the curve;

:. y + dy = f (x + dx) = y + d ̧y. Sx;

Now Sa, Sy are the virtual velocities of the particle for the two forces X, Y;

.. Xdx+ Ydy = 0 by the principle;

.. Xdx+ Ydydx = 0 for all values of dx,

and.. X+ Ydy = 0,

which is the condition of equilibrium.

116. A particle rests upon a smooth curve surface acted on by three forces X, Y, Z parallel to the co-ordinate axes: to find the conditions of equilibrium.

Let x = f(x, y) be the equation of the curve surface, x, y, ≈ being the co-ordinates of the position of equilibrium of the particle. Then if x + dx, y + dy, ≈ + d≈ be the coordinates of the position after disturbance, da, dy, dx are the virtual velocities of the particle for the forces X, Y, Z respectively; and therefore by the principle of virtual velocities,

But because ≈ + dx, y + dy, ≈ + ds are a point in the curve, ≈ + dx = f (x + dx, y + dy)

=≈ + d ̧x. Sx + dy≈ .dy;

.. dx = d ̧x. Sx + dy≈. Sy.

By substituting this value of Sx, we have

(X + Zd ̧≈) dx + (Y + Zd,x) dy = 0.

There is no geometrical relation existing between dy and dx; consequently, the equations of equilibrium are

X + Zd x = 0, Y + Zdy≈ = 0.

117. If two forces P, P' whose virtual velocities are Sp, Sp', act upon a rigid body at different points, and be such that the equation Pdp + P'op' = 0 is true for all arbitrary displacements of the body, then P and P' are equal and act in the same line in opposite directions.

For the equation shews that Sp and Sp' are always zero together. Now disturb the body in such a way that the point at which P acts may remain stationary; then since

F

the body is rigid, the point on which P' acts must have described a circular arc about the stationary point; and as Sp' = 0, that arc must be perpendicular to the direction in which P acts, therefore P' acts in the direction of a normal to the arc, i. e., in a line passing through the point on which P acts. In the same way it may be shewn that P acts in a line passing through the point at which P' acts; hence they both act in the same line: it will therefore be possible to disturb the body so that Sp and Sp' may be equal in magnitude; and they must have different algebraic signs (Pop+ P'dp' = 0), which can only happen, since the body is rigid, by reason of P and P acting in opposite directions; and therefore P and P' are likewise equal.

118. If the equation (Fds) = 0 be true for all arbitrary displacements of a rigid body under the action of external forces F1, F... there is equilibrium.

For if not, there will be at most two resultants (Art. 84); apply forces P, P' equal to these resultants and in the contrary directions to them, and then the body is in equilibrium under the action of the forces F1, F2... P, P'; consequently by the Principle of Virtual Velocities,

Σ(Fds) + Pdp + P'dp' = 0.

But Σ (Fds) = 0 by hypothesis, and therefore Pop + P'op' = 0: and hence it follows from the last article that P and P' are equal and act in opposite directions; consequently they destroy each other; they may therefore be removed without affecting the equilibrium; hence the body is in equilibrium when F1, F2, F.... are the only external forces which act on the body.

119. When a system of connected bodies is in equilibrium under the action of external forces, pressures, &c., the equilibrium would not be affected if the connecting joints, cords, &c. were all to become rigid: and hence any force may be transmitted to any point of the system in the line of its action (Art. 21), providing the original point and the new point of application are not situated in independent parts of the system.

120. If the equation (Fds) =0 be true for all arbitrary displacements of a system of connected rigid bodies, there is equilibrium.

If the system consist of independent parts, let one of those parts alone be displaced, then for that part Σ(Fds) = 0 by hypothesis. If that part is not in equilibrium we may apply forces to each body of it which shall keep each of them in equilibrium: these forces (Art. 119) may be transmitted and reduced to two P, P' acting upon the part under consideration. Hence reasoning as in Art. 118, we find P and P' equal and opposite, and therefore they may be removed without disturbing the equilibrium of the part. The same may be proved of each of the independent parts; and, consequently, the whole system is in equilibrium.

REMARK. We have seen that the principle of virtual velocities is true only when the displacements are so small as to allow us to consider an arc as coincident with its chord or tangent. Now the reader who is familiar with the Differential Calculus will know, that an arc and its tangent coincide analytically only as far as the second term of Taylor's theorem inclusive: hence the principle of virtual velocities embraces only quantities of the first order of smallness. The second term of Taylor's theorem has been called the differential of the first term; wherefore, in applying the principle of virtual velocities, we ought always to use ds instead of ds. The equation of virtual velocities in its proper form is (Fds) = 0. Also because this equation involves only differentials of the first order, it is a matter of indifference whether a body rest upon a curve or its tangent, a surface or its tangent plane; or on any other curve or surface having the same tangent or tangent plane at the point on which it rests.

CHAPTER VI.

ON THE CENTRE OF PARALLEL FORCES, AND ON THE
CENTRE OF GRAVITY.

THE CENTRE OF PARALLEL FORCES.

121. Ir a rigid body be acted on at different points by forces in parallel directions, there is a certain point through which their resultant passes, whatever be the position of the body with respect to the direction in which the forces

act.

Let F1, F2...F, act on the points A, B...K (fig. 25) of a rigid body. From any point O in the body draw the rectangular co-ordinate axes Ox, Oy, Oz. Join A, B; and let the resultant of F1, F2, pass through C. Draw Дa, Bb, Ce parallel to Ox; join a, b passing through c.

Let x1 y11, X2 Y2 2... Yn be the co-ordinates of the points on which the forces act; x'y' those of C; and let be the inclination of AB to ab.

Then

AC sin 0,

Cc BC sin 0;

F、

F2

=

(by Art. 40);

whence we find (F1 + F2) ′ = F1≈1 + F2≈2.

2

Again, take away the forces F, F, and replace them by their resultant F1+ F2 acting at C, then if we put x"y"

2

"

for the co-ordinates of the point through which the resultant of F1, F2, F3, or, which is the same, of the two (F1 + F2) and F, passes, we have as before

(F1 + F2 + F3) ≈′′ = (F1 + F2) ≈′ + F3≈3

2

In this manner, introducing successively one force at a time, until all have been taken in, and denoting by ≈

the co