VOL. XLII.] PHILOSOPHICAL TRANSACTIONS. 643 



tical ; and, the time being given, the gravity may be measured by the space 

 which is the subtense of the angle of contact. In other cases, when the gra- 

 vity varies, or its direction changes, it may be measured at any point by the 

 subtense of the angle of contact, that would have been generated in a given 

 time, if the gravity had continued to act uniformly in parallel lines from that 

 term, that is, by the subtense of the angle of contact in the parabola that has 

 its diameter in the direction of the force, and has the closest contact with the 

 curve; which leads us to the same theorem as before. 



In general, let the gravity, that results from the composition of any number 

 of centripetal forces, which are supposed to act on the body in one plane, be 

 resolved into a force parallel to the ordinates, and a force parallel to the base; 

 then the former shall be measured by the second fluxion of the ordinate, and 

 the latter by the second fluxion of the base, the time being supposed to flow 

 uniformly, so that the velocity of the body may be measured by the fluxion of 

 the curve. When the trajectory is not in one plane, the force is resolved in a 

 similar manner into three forces, which are measured by three second fluxions 

 analogous to them. 



Whether the body move in a void, or in a medium that resists its motion; 

 the gravity that results from the composition of the centripetal forces which 

 act upon the body, is always as the square of its velocity directly, and the 

 chord of the circle of curvature that is in the direction of the gravity in- 

 versely. 



When a body describes any trajectory in a void or in a medium, by a force 

 directed to one given centre, the velocity at any point of the trajectory, is to 

 the velocity by which a circle could be described in a void about the same centre, 

 at the same distance, by the same gravity, in the subduplicate ratio of the 

 angular motion of the ray drawn always from the body to the centre, to the 

 angular motion of the tangent of the trajectory; and, if there be no resistance, 

 the velocity in the trajectory at any point, is the same that would be acquired 

 by the body, if it was to fall from that point through one-fourth of the chord 

 of the circle of curvature that is in the direction of the gravity, and the gra- 

 vity at that point was to be continued uniformly during its descent. 



If the centripetal force be inversely as any power of the distance whose expo- 

 nent is any number m greater than unit, there is a certain velocity (viz. that 

 which is to the velocity in a circle at the same distance as \/ 2 to Vm — 1) which 

 would be just sufficient to carry off' the body upwards in a vertical line, so as 

 that it should continue to ascend for ever, and never return towards the centre. 

 If the body be projecte<i in any other direction with the same velocity, it will 

 describe a trajectory which is here constructed; it is a parabola when m=1, a. 



4n 1 



