OF CENTRAL FORCES. 



31 



For the squares of the tiines are as the distances directly, 

 and as the forces inversely (24a) ; that is, in this case, as the 

 distances and as the squares of the distances, or as the cubes 

 of the distances. 



244. Theorem. The right line joining 

 a revolving body and its centre of attraction, 

 always describes equal areas in equal times, 

 and the velocity of the body is inversely as 

 the perpendicular drawn from the centre to 

 the tangent. 



Let AB be a tangent 

 to any curve in which 

 a body is retained by an 

 attractive force directed 

 to C, and let AB repre- 

 sent its velocity at A, 

 or the space which would be described in an instant of 

 time without disturbance, and AD 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 ; AEG, ECF ; EOF, EGG, being between the 

 same parallels, are equal (117); and if they be infinitely 

 diminished, and the action of C become continual, they will 

 be the evanescent increments of the area described by the 

 revolving radius, while the body moves in the curvilinear 

 orbit ; and the whole areas described in equal times will 

 therefore be equal. And since the constant area ABC^ 



AB.iCH (117, 114), AB=2ABC.— , therefore AB, re- 

 presenting the velocity, is always inversely as CH, or 



1 



u 



245. Theorem. Two bodies being at- 

 tracted towards a given centre, with equal 

 forces, at equal distances, if their velocities 

 be once equal at equal distances, they will 

 remain always equal at equal distances, what- 

 ever be their directions. 



Let one of the bodies descend in the right line AB, 

 towards C, and let the other describe the curve AD, and 

 let the velocities at B and D be equal ; let DE in the tan- 

 gent of AD be the space which would be described in an 



evanescent portion of time by the ve- 

 locity at D, FG the arc of a circle on 

 the centre C, and GE its tangent; and 

 while BF would be described by the 

 velocity at B, let FH be added to it by 

 the attractive force ; draw the arc HI 

 and its tangent IK, and EL HDC, and 

 KL perpendicular to DK, then DG : 

 DE : : GI : EK : : EK : EL, by si- 

 milar triangles ; therefore, GI is to EL 

 in the duplicate ratio of DG to DE, 

 or as the square of DG to the square 

 of DE (124) : therefore EL will be 

 the space described by the attractive force, while DB 

 would be described by the velocity at D ; for the force 

 may be considered as uniform during the description of 

 the evanescent increments ; and the spaces described by 

 means of such a force are as the squares of the times : hence 

 the joint result will be DL, which is ultimately equal to 

 DK, and the whole velocity will be increased in the ratio 

 of DK to DE, or DI to DG, or BH to BF ; consequently, 

 since H, I, and K, are ultimately equidistant from C, the 

 velocities in AB and AD, being always equally increased 

 at equal distances, will therefore always remain equal at 

 equal distances. 



246. Theorem. If a body revolves in 

 an elliptic orbit, by a force directed to one 

 of the foci, the force is inversely as the 

 square of the distance. 



The force is directly as the square of the velocity, and 

 inversely as the deflective chord ; but the velocity is in- 

 versely as the perpendicular falling on the tangent ; there- 

 fore the force is inversely in the joint ratio of the square of 

 the perpendicular and of the deflective chord ; now in the 

 ellipsis, the focal chord varies directly as the square of the 

 distance, and inversely as the square rrf the perpcndiculai 

 {201), consequently this joint ratio is that of the square of' 

 the distance, and the force is always inversely as the square 

 of the distance. 



247. Theorem. Tlie velocity of a body 

 revolving in an ellipsis is equal, at its uaean 

 distance, to the velocity of a body revolving 

 at the same distance in a circle; and the 

 whole times of revolution are equal. 



For the focal chord of curvature at the meaa distance 



