(158.) It has been already stated, that when the body is perfectly free, the centre of gravity must necessarily move downwards, in a direction perpendicular to an horizontal plane. When the body is not free, the circumstances which restrain it generally permit the centre of gravity to move in certain directions, but obstruct its motion in others. Thus if a body be suspended from a fixed point by a flexible cord, the centre of gravity is free to move in every direction except those which would carry it farther from the point of suspension than the length of the cord. Hence if we conceive a globe or sphere to surround the point of suspension on every side to a distance equal to that of the centre of gravity from the point of suspension, when the cord is fully stretched, the centre of gravity will be at liberty to move in every direction within this sphere. There are an infinite variety of circumstances under which the motion of a body may be restrained, and in which a most important and useful class of mechanical problems originate. Before we notice others, we shall, however, examine that which has just been described more particularly. Let P, _fig. 44._, be the point of suspension, and C the centre of gravity, and suppose the body so placed that C shall be within the sphere already described. The cord will therefore be slackened, and in this state the body will be free. The centre of gravity will therefore descend in the perpendicular direction until the cord becomes fully extended; the tension will then prevent its further motion in the perpendicular direction. The downward force must now be considered as the diagonal of a parallelogram, and equivalent to two forces C D and C E, in the directions of the sides, as already explained in (149). The force C D will bring the centre of gravity into the direction P F, perpendicularly under the point of suspension. Since the force of gravity acts continually on C in its approach to P F, it will move towards that line with accelerated speed, and when it has arrived there it will have acquired a force to which no obstruction is immediately opposed, and consequently by its inertia it retains this force, and moves beyond P F on the other side. But when the point C gets into the line P F, it is in the lowest possible position; for it is at the lowest point of the sphere which limits its motion. When it passes to the other side of P F, it must therefore begin to ascend, and the force of gravity, which, in the former case, accelerated its descent, will now for the same reason, and with equal energy, oppose its ascent. This will be easily understood. Let C′ be any point which it may have attained in ascending; C′ G′, the force of gravity, is now equivalent to C′ D′ and C′ E′. The latter as before produces tension; but the former C′ D′ is in a direction immediately opposed to the motion, and therefore retards it. This retardation will continue until all the