where a is some constant quantity for all couples. If a couple whose moment equal 1, be taken to equal 1, or be taken for the unit of couple in terms of which all others shall be measured, we must have from above

1 = a,

therefore on this supposition

C = M,

or a couple is always represented by its moment. The unit of couple spoken of may very well be the couple whose force is equal to the unit of force and whose arm is equal to the unit of length, for such a couple would necessarily have its moment equal 1.

70. If three forces act upon a point in space, and lines be drawn from that point parallel and proportional to them, their resultant is parallel and proportional to the diagonal of the parallelopiped described upon these three lines as edges.

Let the three forces P, Q, R act upon the point A, and

let the lines AB, AC, AD be taken parallel and proportional to them respectively. Upon these three lines as edges complete the parallelopiped, ACEFB; then the resultant of P, Q and R acts along, and is in the same proportion to AF, as P, Q, and R are to AB, AC, AD respectively.

Join AC, BF.

Because AE is the diagonal of the parallelogram described upon AC, AD, it is in the direction of and proportional to the resultant of Q and R.

Again, because AF is the diagonal of the parallelogram described upon AE, AB, it is parallel and proportional to the resultant of the forces represented by them, i. e. it represents in magnitude and direction the resultant of the force P and the resultant of Q and R: it therefore represents the resultant of the three given forces.

71. The results of Articles (32) and (37) may be included in the following general enunciation.

If a body be in contact with a smooth plane at any number of points, and these points be joined successively so as to form a polygonal figure, since the resistances of the plane upon the body at these points form a system of parallel forces acting in the same direction, their resultant will evidently be parallel to them, and will pass through some point within the polygonal figure. Hence for equilibrium the resultant of the other forces acting upon the body must be perpendicular to and act towards the plane, and must pass through some point within the above-mentioned polygonal figure. It is indifferent whereabouts within the figure this point be situated, for the resistances at the points of contact are indeterminate and may be exerted to any required amount; and in all cases just so much force will be called into action at each point as will make the resultant resistance act at the same point as the resultant of the other forces; only this is not possible when the latter does not fulfil the condition of passing within the polygonal figure.

If the resultant of the other forces be perpendicular to the plane, but fall without the figure of contact, the body will turn over. If it be not perpendicular to the plane, but yet pass through the figure, the body will slide. If it be neither perpendicular nor yet pass through the figure, the body will begin to both slide and turn over.

72. When gravity is the only force acting upon the body besides the resistances of the plane, the foregoing proposition reduces itself to the assertion that the vertical line through the centre of gravity of the body must be perpendicular to the plane, and must not fall withoutside the polygonal figure formed by joining the successive points of contact.

SECTION II.

VIRTUAL VELOCITIES.

73. A large class of Statical Problems may be easily solved by the aid of an artifice which is termed the Principle of Virtual Velocities. Although it does not strictly belong to geometrical Statics, it deserves to be mentioned on account of some of the remarkable results to which it leads us.

DEF. Suppose any number of forces to be acting at different points of a body, and suppose the body to receive an indefinitely small displacement: if now perpendiculars be drawn from the new positions of the points of application of the forces upon the directions of the forces as they were before the displacement, the line intercepted between the foot of any one perpendicular and the first position of the point of application of the corresponding force is called the virtual velocity of that force, and is estimated as positive or negative according as it falls on the side of the point towards which the force acts, or the contrary.

74. The Principle of Virtual Velocities asserts that if any number of external forces acting upon a body or system of points, be in equilibrium, then the algebraical sum of the terms formed by multiplying each force by its virtual velocity vanishes thus if P,P,.... P, represent the forces, P1PP their respective virtual velocities when the body has received any given indefinitely small displacement whatever, if P‚P„...P ̧ are in equilibrium with each they must satisfy other the relation

P1P1 + P2P2 + + Pmln = 0..

....

....... •

(1).

n

n

It is difficult, or perhaps impossible, to give a general proof of this assertion, based solely upon the properties of force and independent of its particular mode of action; but its truth has

been ascertained for almost every conceivable arrangement or method by which forces may act upon a system of points: a few only of the simplest cases will be given here in which it is verified, and a few examples will be solved by its application.

75. If the supposed displacement be made in such a manner that some of the virtual velocities equal zero, i.e. that some of the p's in equation (1) vanish, the same number of terms, with their corresponding forces, will disappear from the equation: the analytical relation which the remaining forces must satisfy will therefore be simplified. In using equation (1) it would always be our object to make the displacement so as to get rid of the forces whose value we do not care to find: theoretically there is no difficulty in doing this, but in practice all displacements do not afford equal facilities for finding the geometrical quantities PP2.... P; we are obliged to choose those which are most convenient. It is therefore readily seen that the Principle of Virtual Velocities is most useful in those cases where the simplest displacement at the same time retains only those forces which we desire chiefly to consider: such happens when some of the forces are supplied by contact with smooth surfaces; for if the displacement be made by sliding the system along them, the virtual velocities of the reactions are evidently nothing, and the forces themselves disappear from the above relation.

76. If PP,....P be any number of forces acting in the same plane at the same point, R their resultant, PP2....Pr their respective virtual velocities consequent upon a displacement of the point made in that plane; then, generally,

P1P1+ P2P2+ ... + PP2 = Rr.

2

2

First let us consider only the first two forces as PP, acting upon the point A, and let AB, AC represent them in magnitude and direction; their resultant R, will be represented by AD the diagonal of the parallelogram described upon AB, AC. Suppose A to be displaced to a position A', and join AA'.