By successive applications of the above rule any number of forces acting on a particle may be replaced by a single force which is the vector-sum of the given forces: this single force is called the _resultant_. Thus if [->AB], [->BC], [->CD] ..., [->HK] be vectors representing the given forces, the resultant will be given by [->AK]. It will be understood that the figure ABCD ... K need not be confined to one plane. [Illustration: FIG. 2.] If, in particular, the point K coincides with A, so that the resultant vanishes, the given system of forces is said to be in _equilibrium_--i.e. the particle could remain permanently at rest under its action. This is the proposition known as the _polygon of forces_. In the particular case of three forces it reduces to the _triangle of forces_, viz. "If three forces acting on a particle are represented as to magnitude and direction by the sides of a triangle taken in order, they are in equilibrium." A sort of converse proposition is frequently useful, viz. if three forces acting on a particle be in equilibrium, and any triangle be constructed whose sides are respectively parallel to the forces, the magnitudes of the forces will be to one another as the corresponding sides of the triangle. This follows from the fact that all such triangles are necessarily similar. [Illustration: FIG. 3.] As a simple example of the geometrical method of treating statical problems we may consider the equilibrium of a particle on a "rough" inclined plane. The usual empirical law of sliding friction is that the mutual action between two plane surfaces in contact, or between a particle and a curve or surface, cannot make with the normal an angle exceeding a certain limit [lambda] called the _angle of friction_. If the conditions of equilibrium require an obliquity greater than this, sliding will take place. The precise value of [lambda] will vary with the nature and condition of the surfaces in contact. In the case of a body simply resting on an inclined plane, the reaction must of course be vertical, for equilibrium, and the slope [alpha] of the plane must therefore not exceed [lambda]. For this reason [lambda] is also known as the _angle of repose_. If [alpha] > [lambda], a force P must be applied in order to maintain equilibrium; let [theta] be the inclination of P to the plane, as shown in the left-hand diagram. The relations between this force P, the gravity W of the body, and the reaction S of the plane are then determined by a triangle of forces HKL. Since the inclination of S to the normal cannot exceed [lambda] on either side, the value of P must lie between two limits which are represented by L1H, L2H, in the right-hand diagram. Denoting these limits by P1, P2, we have P1/W = L1H/HK = sin ([alpha] - [lambda])/cos ([theta] + [lambda]), P2/W = L2H/HK = sin ([alpha] + [lambda])/cos ([theta] - [lambda]).