CAB

From this we infer that if a point A be kept in equilibrium by the action of three forces P, Q, and R, each of these forces is proportional to the sine of the angle formed by the direction of the two others. It is also a consequence of this proposition that if three forces be in equilibrium they must all act in the same plane, and any two of them must be greater than the third. The resultant of three forces X, Y, Z, applied to the same point A in space, and severally represented in magnitude and direction by the straight lines A B, A C, A D, is represented by A F, the diagonal of the parallelepiped on, of which the sides are A B, A C, and A D. For the two forces X and Y, which are represented by A B and • A C, the two sides of the parallelogram A B G C will have for their resultant a force P, represented by A G, the diagonal of this parallelogram. And because A D is equal and parallel to G F, the figure A D F G is a parallelogram; and consequently the two forces P and Z, represented by A G and A D, will have for their resultant a force R, represented by A F, the diagonal of the parallelogram A D F G, which is also the diagonal of the parallelepiped on. From this theorem it follows that any force whatever R can always be decomposed into three others, X, Y, Z, respectively parallel to three straight lines given in space, provided that no two of them be parallel; and if each of the three given lines be at right angles to the plane of the other two, and the angles which A F (the direction of the given force R) makes respectively with A B, A C, A D (the directions of the forces X, Y, Z) be denoted by a, b, c, we have then, evidently, X = R COS. a, Y = R cos. 6, Z = R cos. c, the three angles a, b, c, being connected by the relation C0S.2o + cos. 25 -I- [s. 475]