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From tl as we infer that if a point A be kept in equilibrium by the action of three forces P, Q, and R, each of these forcesisproiKirtional to the sine of the angle formed by tire 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..B 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 equ;U and parallel toG 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. b, Z=R cos. c, the three angles a, b, c, being connected by the relation cos. 2a+cos. 2i-|-coe. 2C=1. In t>.3se investigations in mechanics where a number of forces are concerned, it is usual to resolve them all into three forces parallel to three rectangular co-ordinates; when the resultant of the three sets of rectangular forces will evidently be the common resultant of all the forces. If F, F', F", F'", . be the forces; a, a', a", a'", &,c. the angles which they make with one of the three a.xes; b, b', b", h'", . the angles which they make with another of the axes; and c, c', c", c'", &.C. the angles which they make with the third; and X, Y, Z be the three rectangular forces which are the sums of the components of all the original forces, F, F', F", F'", then, X=F cos. a+F' cos. a'+F" cos. a"-\- F" cos. a'"-|- . Y=F cos. 4-j-F' cos. 4'4-F" cos. &"-4-F"' cos. 6"'+ . Z=F cos. c+F' cos. c'-\-F" cos. e"-j-F'" cos. c '-f- . As the three forces X, Y, Z are not situated in the same plane, and can therefore never be in equilibrium so long as any one of them has a real value, in order that the body may remain at rest under the action of all the given forces F, F", F", F ", , it is nece ary that the three conditions be fulfilled; namely, X=0, Y=0, Z=;0. Forces have different denominations according to the manner in which they act; thus we have Accelerating Forces, Central Forces, Parallel Forces, Uniform and Variable Forces, . See Acceleration, Central Force, Pre ure, . Accelerating Force. — An accelerating force is that which continues to act upon a body after it has been put in motion;whence the body moves with a variable velocity, and, when the intensity of the force is constant, receives equal increments of velocity in equal intervals of time. We have a familiar example of an accelerating force of this kind in terrestial gravity, under the action of which heavy bodies dropped from a height fall to the ground with a constantly accelerated velocity. When a body is urged by an accelerating force, the relations bstween the force, the velocity, the space pa ed over, and the time, are expre ed by three equations, which are called the equations of motion, and may be investigated as follows: Let t denote the time from the commencement of the motion, v the velocity acquired at the end of the time t, and t the space described by the moving body. During an inf i- 466 [s. 480]