394 PHILOSOPHICAL TRANSACTIONS. [anNO 1788. 



find the sum f and p' of the forces at the point p in directions which are not 

 both situated in one plane with the tangent and each other; and also the chords 

 c and c' of curvature in those directions in terms of the distances from 3 points, 

 or 2 abscissae and 1 ordinate, &c. In the equations v^ = -^f X c and v^ = -ip' 

 X c' for v^ substitute its value found from the principles before given ; and there 

 result 2 fluxional equations of the 2d order, expressing the relation between the 

 distances from 3 points, or 2 abscissae and an ordinate, &c. 



Prop. 8. — Fig. 11. Let a body move in a curve vp, &c. and be acted on at p' 

 by a force /, (which is as any function of the distance sp') tending to s; let the 

 velocities at p and p be represented by the, lines yp. and yp in the direction of the 

 tangents to the points p and jb; resolve these forces yp and yp into 2 others Yk 

 and kv, and yl and Ip, of which one kr and yl is parallel to the line sl; the 

 other k-p and Ip is parallel to mp ; let a body fall in the right line ls, and the 

 force acting on the body at m' be to the force acting on the body moving in the 

 curve at p' :: sm' : sp', and p'm', pm and pm be perpendicular to sl; then if the 

 velocity of the body falling in the right line sl at the point m be kr, the velocity 

 of the body at the point m acted on by the above-mentioned forces will be yl. 

 This is easily demonstrated from the resolution of forces. 



2. Through s draw sn parallel to pm or pm, &c., and assume in the line (sn) 

 SP = PM and sp :=pm, and let the force at p' in the line sn and distance = m'p': 

 the force of the body moving in the curve at the distance p's :: p'm' : sp'; then 

 if the velocity at the distance sp = pm be p^^, the velocity at the distance 

 sp = pm will be pi. 



Cor. The force in the direction of the line sl vanishes in the point where a 

 perpendicular sn to the line sl passing through the point s cuts the curve, and 

 consequently the velocity in the direction of sl in that point is the greatest or 

 least, &c.; but if the tangent of the curve be perpendicular in any point to 

 LS, then the velocity in the direction ls is nothing: the same may be applied to 

 the velocity in any other direction. 



Exam. Fig. 12. Let a body move in the circumference of a circle spa, of 

 which the centre of force is a point s in the circumference; it is known that the 

 force in the direction and at the distance sp is as sp~*; but the force in the 

 direction sp is by the hypothesis to the direction (sa) :: sp : sm, if pm be per- 

 pendicular to SM, and consequently the force in the direction (sa) is as sm x 

 sp~®; but if as be a diameter, as X sm = sp*; therefore smx sp~^ = sm x 

 as"-* X sm~' = '• — Y ; and the diameter as being given, the force in the line sa 

 varies as sm~*, that is, inversely as the square of the distance: if the force 

 varies as sm~*= a?~^, then -ui will vary as ——^, where v denotes the velocity; and 



V* will vary as - — — , which agrees with the square of the velocity deduced 



