former depends on the figure and density of the body, and the velocity of its motion. These very complex effects render a simple and elementary exposition of the mechanical properties of a fixed axis a matter of considerable difficulty. Indeed, the complete mathematical development of this theory long eluded the skill of the most acute geometers, and it was only at a comparatively late period that it yielded to the searching analysis of modern science. (182.) To commence with the most simple case, we shall consider the body as submitted to the action of a single force. The effect of this force will vary according to the relation of its direction to that of the axis. There are two ways in which a body may be conceived to be moveable around an axis. 1. By having pivots at two points which rest in sockets, so that when the body is moved it must revolve round the right line joining the pivots as an axis. 2. A thin cylindrical rod may pass through the body, on which it may turn in the same manner as a wheel upon its axle. If the force be applied to the body in the direction of the axis, it is evident that no motion can ensue, and the effect produced will be a pressure on that pivot towards which the force is directed. If in this case the body revolved on a cylindrical rod, the tendency of the force would be to make it slide along the rod without revolving round it. Let us next suppose the force to be applied not in the direction of the axis itself, but parallel to it. Let A B, _fig. 70._, be the axis, and let C D be the direction of the force applied. The pivots being supposed to be at A and B, draw A G and B F perpendicular to A B. The force C D will be equivalent to three forces, one acting from B towards A, equal in quantity to the force C D. This force will evidently produce a corresponding pressure on the pivot A. The other two forces will act in the directions A G and B F, and will have respectively to the force C D the same proportion as A E has to A B. Such will be the mechanical effect of a force C D parallel to the axis. And as these effects are all directed on the pivots, no motion can ensue. If the body revolve on a cylindrical rod, the forces A G and B F would produce a strain upon the axis, while the third force in the direction B A would have a tendency to make the body slide along it. (183.) If the force applied to the body be directed upon the axis, and at right angles to it, no motion can be produced. In this case, if the body be supported by pivots at A and B, the force K L, perpendicular to the line A B, will be distributed between the pivots, producing a pressure on each proportional to its distance from the other. The pressure on A having to the pressure on B the same proportion as L B has to L A.