string are attached. Supposing the increase of the length of the string, when stretched in a straight line by a force equal to the weight of the rod, to be equal to twice the length of the rod, to determine the position of equilibrium under the present circumstances.

=

If 2a = the natural length of the string, 2b the length of the rod, and the angle included between the two parts of the string,

=

(11) Two equal rigid rods AC, BC, without weight, are connected together by a smooth hinge at C and rest in a vertical plane, their lower extremities, which are tied together by an elastic string AB, being placed upon a smooth horizontal plane. If a be the inclination of each rod to the horizon, when a weight W is fixed to the middle point of each, and a the inclination when a weight W' is so fixed; to find the natural length of the string.

If a be the length of each rod, the natural length of the string is equal to

(12) Two fine strings, slightly elastic, are fastened to the middle points of the sides of a uniform rectangular board, thus crossing the board parallel to its sides, and intersecting in the centre. Supposing the board to be suspended from the intersection of the strings, to find approximately the distance at which it will hang below the point of suspension.

If W be the weight of the board, 2a, 2b, the lengths of its sides, and λ the modulus of the elasticity of the strings, the required distance is equal to

(13) Six equal rods are connected together by hinges at their ends so as to form a hexagon, and, one of the rods being supported in a horizontal position, the opposite one is fastened to it by a fine elastic string joining their middle points. Supposing the modulus of elasticity to be equal to the weight of each rod, to find the natural length of the string in order that the hexagon may be equiangular in the position of equilibrium. If a the length of each rod, and 7= the natural length of the string,

√3
l= a.
4

(14) A heavy elastic string is laid upon a smooth double inclined plane in such a manner as to remain at rest: to find how much the string is stretched.

If W= the weight of the string, c = its natural length, and α, a, denote the inclinations of the planes; then the required extension is equal to

(15) An elastic string, the upper end of which is fixed, rests on a rough inclined plane in the direction of greatest slope: to determine the limits of the extension of the string beyond its natural length.

Let a be the inclination of the plane; I the natural length of the string, and l' that of a portion of it the weight of which is equal to its modulus of elasticity; e the angle of indifference: then the extension of the string will lie between the limits.

CHAPTER VI.

VIRTUAL VELOCITIES.

THE Principle of Virtual Velocities consists in the following general proposition :

"If a material system, acted on by any forces whatever, be in equilibrium; and we conceive this system to experience, consistently with its geometrical relations, any indefinitely small arbitrary displacement; the sum of the forces, multiplied each of them by the resolved part, parallel to its direction, of the space described by its point of application, will be equal to zero; this resolved part being considered positive when it lies in the direction of its corresponding force, and negative when in an opposite direction."

The resolved parts of the spaces described by the points of application of the forces are called their Virtual Velocities. Let P, Q, R,... denote any system of forces acting on a system of points consistently with equilibrium; and let a, B, y,..... denote their virtual velocities; then, as far as the first powers of a, B, Y,...... are concerned,

Pa + QB + Ry + S8+......0.........(A).

The Principle of Virtual Velocities was first detected by Guido Ubaldi1 as a property of the equilibrium of the lever and of moveable pullies. Its existence was afterwards recognized by Galileo' in the inclined plane, and the machines depending upon it. The expression 'moment' of a force or weight acting on any machine, was used by Galileo to denote its energy or effort to set the machine in motion, who accordingly declared that for the equilibrium of a machine, acted on by two forces,

1 Mechanicorum Liber; De Libra, De Cochlea.

2 Della Scienza Mecanica, Opera, Tom. 1. p. 265; Bologna, 1655.

it is necessary that their moments should be equal, and should take place in opposite directions; he shewed moreover that the moment of a force is always proportional to the force multiplied by its virtual velocity. The word 'moment' was used in the same sense by Wallis', who adopted Galileo's principle of the equality of moments as the fundamental principle of Statics; and deduced from it the conditions for the equilibrium of the principal machines. Descartes' has likewise reduced the whole science of Statics to a single principle, which virtually coincides with that of Galileo; it is presented however under a less general aspect. The principle is, that it requires precisely the same force to raise a weight P through an altitude a, as a weight through an altitude b, provided that P is to Q as b to a. From this it follows, that two weights attached to a machine will be in equilibrium when they are disposed in such a manner that the small vertical paths which they can simultaneously describe are reciprocally as the weights.

Torricelli is the author of another principle which may be immediately deduced from the principle of virtual velocities: the principle is, that when any two weights rigidly connected together are so placed that their centre of gravity is in the lowest position which it can assume consistently with the geometrical conditions to which they are subject, they will be in equilibrium. The principle of Torricelli has given birth to the following more general one, viz.-that any system whatever of heavy bodies will be in equilibrium when their centre of gravity is in its lowest or highest position.

John Bernoulli was the first to announce the principle of virtual velocities under its most general aspect in the form which we have given above, in a letter to Varignon, dated Bâle, Jan. 26, 1717. The striking value of the principle, as an instrument of analytical generalization, has been splendidly exhibited by Lagrange in his Mécanique Analytique.

1 Mechanica, sive de Motu, Tractatus Geometricus.

2 Lettre 73, Tom. 1. 1657; de Mechanica Tractatus, Opuscula Posthuma. 3 De Motu gravium naturaliter descendentium, 1644.

4 Nouvelle Mécanique, Tom. 11. sect. 9.

From the principle of virtual velocities may be immediately deduced the principle which was proposed by Maupertuis in the Mémoires de l'Académie des Sciences de Paris for the year 1740, under the name of the Loi de Repos; and which Euler has developed at large in the Mémoires de l'Académie de Berlin for the year 1751. Suppose that any number of forces P, Q, R,... tending towards fixed centres and functional of their distances p, q, r,... from the centres, act on a system of points rigidly connected together. Then, supposing the system of points to be slightly displaced, so that p, q, r,... receive increments dp, dq, dr,...we shall have, by the principle of virtual velocities,

Pdp+Qdq+ Rdr+... = 0.

Let dII denote the left-hand member of this equation; then dII = 0.........(B).

From this it appears that, if the system be so placed that II may have a maximum or a minimum value, there will be equilibrium: this proposition constitutes Maupertuis' Principle of Rest. It does not however follow conversely that, whenever the system is at rest, II shall have a maximum or minimum value, since by the principles of the differential calculus we know that the equation (B), although a necessary, is not the only condition for the existence of such a value. Lagrange1 has shewn that, if II be a minimum, the equilibrium will be stable, and, if a maximum, unstable.

As an example of this theory, it is evident that, if any system be in equilibrium under the action of gravity, there will be stable or unstable equilibrium accordingly as the centre of gravity is in the lowest or highest position which is compatible with the geometrical relations to which the system is subject.

The principle of equilibrium developed by Courtivron is likewise grounded upon the principle of virtual velocities; Courtivron's Principle asserts that, if a system of bodies be in motion under the action of any forces varying according to any

1 Mécanique Analytique. Première Partie, sect. 5.

2 Mémoires de l'Académie des Sciences de Berlin, 1748, 1749.