(82.) It is frequently necessary to express the portion of a given force, which acts in some given direction different from the immediate direction of the force itself. Thus, if a force act from A, _fig. 11._, in the direction A C, we may require to estimate what part of that force acts in the direction A B. If the force be a pressure, take as many inches A P from A, on the line A C, as there are ounces in the force, and from P draw P M perpendicular to A B; then the part of the force which acts along A B will be as many ounces as there are inches in A M. The force A B is mechanically equivalent to two forces, expressed by the sides A M and A N of the parallelogram; but A N, being perpendicular to A B, can have no effect on a body at A, in the direction of A B, and therefore the effective part of the force A P in the direction A B is expressed by A M. (83.) Any number of forces acting on the same point of a body may be replaced by a single force, which is mechanically equivalent to them, and which is, therefore, their resultant. This composition may be effected by the successive application of the parallelogram of forces. Let the several forces be called A, B, C, D, E, &c. Draw the parallelogram whose sides express the forces A and B, and let its diagonal be A′. The force expressed by A′ will be equivalent to A and B. Then draw the parallelogram whose sides express the forces A′ and C, and let its diagonal be B′. This diagonal will express a force mechanically equivalent to A′ and C. But A′ is mechanically equivalent to A and B, and therefore B′ is mechanically equivalent to A, B, and C. Next construct a parallelogram, whose sides express the forces B′ and D, and let its diagonal be C′. The force expressed by C′ will be mechanically equivalent to the forces B′ and D; but the force B′ is equivalent to A, B, C, and therefore C′ is equivalent to A, B, C, and D. By continuing this process it is evident, that a single force may be found, which will be equivalent to, and may be always substituted for, any number of forces which act upon the same point. If the forces which act upon the point neutralise each other, so that no motion can ensue, they are said to be in equilibrium. (84.) Examples of the composition of motion and pressure are continually presenting themselves. They occur in almost every instance of motion or force which falls under our observation. The difficulty is to find an example which, strictly speaking, is a simple motion. When a boat is rowed across a river, in which there is a current, it will not move in the direction in which it is impelled by the oars. Neither will it take the direction of the stream, but will proceed exactly in that intermediate direction which is determined by the composition of force.