228 CENTRAL FORCES; 



SBC is equal to the triangle S B c ; consequently the motion 

 through AB and BC, or the times, are as the two triangles 



R 



A 



SAB and SBC; and so it may be proved if the force acting 

 towards S again deflects the body at C, making it move in the 

 diagonal C D. If, now, instead of this deflecting force acting 

 at intervals A, B, C, it acts at every instant, the intervals of 

 time become less than any assignable time, and then the 

 spaces A B, B C, CD will become also indefinitely small and 

 numerous, and they will form a curve line ; and the straight 

 lines drawn from any part of that curve to S will describe 

 curvilinear areas, as the body moves in the curve A B C D, 

 those areas being proportional to the times. So conversely, 

 if the triangles S B c and SBC are equal, they are between 

 the same parallels, and c C is parallel to S B, and D d to S C ; 

 consequently the force which deflects acts in the lines S B and 

 S C, or towards the point S. It is equally manifest that the 

 direction of the lines Be, C d, from which the centripetal 

 force deflects the body, is that of tangents to the curve which 

 the body describes, and that consequently the velocity of the 

 body is in any given point inversely proportional to the 

 perpendicular drawn from the centre to the tangent ; the 

 areas of the triangles whose bases are equal, being in the pro- 

 portion of their altitude, that is, of those perpendiculars, and 

 those areas being by the proposition, proportional to the times. 

 There are several other corollaries to this important pro- 

 position which deserve particular attention. B c and D e are 



