OF DEFLECTIVE FORCES, 113 



262, Corollary 4. " 243/' If the 

 forces are inversely as the squares of the 

 distances, the squares of the times are as the 

 cubes of the distances. 



For the squares of the times are as the distances di- 

 rectly, and as the forces inversely (261): that is, in this 

 case, as the distances, and as the squares of the distances, 

 or as the cubes of the distances. 



263. Theorem. " 244.'' The right line, 

 joining a revolving body and its centre of 

 attraction, always describes equal areas in 

 equal times, and the velocity oT ths body is 

 inversely as the perpendicular drawn from the 

 centre to the tangent. 



Let AB be a tangent to any jj 

 curve, in which a body is retained 

 by an attractive force directed to 

 C, and let AB lepresent its velo- 

 city at A, or the space which C 

 would be described in an instant of time without distur- 

 bance, and AD the space which would be described by 

 the action of C in the same time ; then completing the 

 parallelogram, AE will be the joint result (226); again, 

 take EF=: AE, and EF will now represent its spontaneous 

 motion in another equal instant of time, and by the action 

 of C it will again describe the diagonal of a parallelogram 

 EG; but the triangles ABC, AEC ; AEC, ECF; ECF, 

 ECG, being between the same parallels, are equal (117); 

 and if these triangles be infinitely diminished, and the 

 action of C become continual, they will be the evanescent 



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