ON DEFLECTIVE FORCES. 37 



rizontal 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 maybe 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 tbe 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 ; that is, that the 

 force directed to its focus must be inversely as the square of the distance. 

 We have no other expcrimentul proof of this theorem than astronomical ob- 

 servations, which are indeed perfectly decisive, but do not require to be here 

 anticipated. (Plate II. Fig. 16.) 



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

 parison of rectilinear and curvilinear motions. Two bodies being attracted 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 re- 

 main always equal at equal distances, whatever be their directions. For in- 

 stance, if one cannon ball be slxot oblitjuely upwards, and anotlier perpendi- 

 cularly 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 lias usually been made a step in tbe demonstra- 

 tion 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 fuom the 

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

 sandth part, or three grains and a half for every pound, and w.e may discover 

 this inequality by means of the vibrations of pendulutns, which become a lit- 



