It appears, moreover, that if [theta] be varied P will be least when L1H is at right angles to KL1, in which case P1 = W sin ([alpha] - [lambda]), corresponding to [theta] = -[lambda]. [Illustration: FIG. 4.] Just as two or more forces can be combined into a single resultant, so a single force may be _resolved_ into _components_ acting in assigned directions. Thus a force can be uniquely resolved into two components acting in two assigned directions in the same plane with it by an inversion of the parallelogram construction of fig. 1. If, as is usually most convenient, the two assigned directions are at right angles, the two components of a force P will be P cos [theta], P sin [theta], where [theta] is the inclination of P to the direction of the former component. This leads to formulae for the analytical reduction of a system of coplanar forces acting on a particle. Adopting rectangular axes Ox, Oy, in the plane of the forces, and distinguishing the various forces of the system by suffixes, we can replace the system by two forces X, Y, in the direction of co-ordinate axes; viz.-- X = P1 cos [theta]1 + P2 cos [theta]2 + ... = [Sigma](P cos [theta]), } Y = P1 sin [theta]1 + P2 sin [theta]2 + ... = [Sigma](P sin [theta]). } (1) These two forces X, Y, may be combined into a single resultant R making an angle [phi] with Ox, provided X = R cos [phi], Y = R sin [phi], (2) whence R² = X² + Y², tan [phi] = Y/X. (3) For equilibrium we must have R = 0, which requires X = 0, Y = 0; in words, the sum of the components of the system must be zero for each of two perpendicular directions in the plane. [Illustration: FIG. 5.] A similar procedure applies to a three-dimensional system. Thus if, O being the origin, [->OH] represent any force P of the system, the planes drawn through H parallel to the co-ordinate planes will enclose with the latter a parallelepiped, and it is evident that [->OH] is the geometric sum of [->OA], [->AN], [->NH], or [->OA], [->OB], [->OC], in the figure. Hence P is equivalent to three forces Pl, Pm, Pn acting along Ox, Oy, Oz, respectively, where l, m, n, are the "direction-ratios" of [->OH]. The whole system can be reduced in this way to three forces X = [Sigma] (Pl), Y = [Sigma] (Pm), Z = [Sigma] (Pn), (4) acting along the co-ordinate axes. These can again be combined into a single resultant R acting in the direction ([lambda], [mu], [nu]), provided X = R[lambda], Y = R[mu], Z = R[nu]. (5) If the axes are rectangular, the direction-ratios become direction-cosines, so that [lambda]² + [mu]² + [nu]² = 1, whence R² = X² + Y² + Z². (6) The conditions of equilibrium are X = 0, Y = 0, Z = 0.