ON DEFLECTIVE FORCES. 29 



either be held in the hand, or have a weight attached to it. The ball being 

 made to revolve, its motion becomes evidently more rapia when it is drawn 

 by the horizontal thread nearer to the fixed point, and slower when it is 

 suffered to fly off to a greater distance. (Plate II. Fig. 15.) 



It was also discovered by Kepler* that each of the planets revolves in an 

 ellipsis, of which the sun occupies one of the foci. It is well known that an 

 ellipsis is an oval figure, which may be described by fixing the ends of a 

 thread to two points, and moving a tracing point so that it may always be 

 at the point of the angle formed by the thread ; and that the two fixed 

 points are called its foci. The inference respecting the force by which a 

 body may be made to revolve in an ellipsis, was first made by Newton ;t 

 that is, that the force directed to its focus must be inversely as the square 

 of the distance. We have no other experimental proof of this theorem than 

 astronomical observations, which are indeed perfectly decisive, but do not 

 i require to be here anticipated. (Plate II. Fig. 16.) 



There is another general proposition which is sometimes of use in the 

 comparison of rectilinear and curvilinear motions. Two bodies being at- 

 tracted in the same manner towards a given centre, that is, with equal 

 forces at equal distances, if their velocities be once equal at equal distances, 

 they will remain always equal at equal distances whatever be their direc- 

 tions. J For instance, if one cannon ball be shot obliquely upwards, and 

 another perpendicularly upwards with the same velocity, the one will 

 describe a curve, and the other a straight line, but their velocities will 

 always remain equal, not at the same instants of time, but at equal distances 

 from the earth's centre, or after having ascended through equal vertical 

 heights, although in different directions. This proposition has usually 

 been made a step in the demonstration of the law of the force by which a 

 body is made to revolve in an ellipsis, but there is a much simpler method 

 of demonstrating that law, by means of some properties of the curvature of 

 the ellipsis. 



In treating of the motion of projectiles, the force of gravitation may, 

 without sensible error, be considered as an equable force, acting in parallel 

 lines perpendicular to the horizon. In reality, if we ascend a mile from 

 the earth's surface, the actual weight of a body is diminished about a two 

 thousandth part, or three grains and a half for every pound ; and we may 

 discover this inequality by means of the vibrations of pendulums, which 

 become a little slower when they are placed on the summits of very high 

 mountains. On the other hand, a body not specifically heavier than water 

 gains more in apparent weight on account of the diminished density of the 

 atmosphere at great elevations, than it loses by the increase of its distance 

 from the earth. But both these differences may, in all common calcula- 

 tions, be wholly disregarded. The direction of gravity is always exactly 

 perpendicular to the horizon, that is, to the surface of the earth, which is 

 somewhat curved, on account of the earth's spheroidical figure ; but any 

 small portion of this surface may be practically considered as a plane, and 

 the vertical lines perpendicular to it as parallel to each other. 



* On the Motion of Mars, c. 58. f Principia, I. 11. 



J Prop. 40. 



