402 PHILOSOPHICAL TRANSACTIONS. [aNNO 1788. 



direction perpendicular to the tangent, and from their fluxions i and ^, and the 

 force / may, by the method before given, be deduced the forces in the 2 direc- 

 tions above-mentioned; and in the same manner may be found, from ar, y, i', 

 and y, the forces in the directions of the tangent and perpendicular to it, which 

 follow from all the forces tending to given points, and acting on the body movmg 

 in the curve to be investigated. 2. If some of the centres of force move in 

 given curves b, b', b'\ &c. whose arcs let be denoted by b, b', &c. and their 

 respective places at the same time are given; then from their respective places 

 given and forces, and a? and z/, and .v and y, can, as before, be deduced the 

 forces in the direction of the tangent and its perpendicular to the curve required. 

 3. If other centres of forces move in given curves a, a', a", &c. and the velo- 

 cities are given at every point of the curves ; let a, a', a'', &c. be the arcs of 

 the curves a, a' a", &c. and suppose Vj v\ v"y &c. their correspondent veloci- 

 ties; then, if the increments of the time be given, v/ill - = ^ = ^ == &c. but 

 as the velocities are given at every point of the curves, v in the curve (a) will be given 

 in terms of its absciss, ordinate, arc, &c. and consequently - in terms of the same 



quantities and their first fluxions; the same may be aflirmed of the fluxions -„ V, 



in the curves a', a", &c. ; hence, from the equation - = -V, can be deduced 

 the relation between the absciss or ordinate, &c. of the curve a and its correspon- 

 dent absciss or ordinate, &c. of the curve a'; and so of the remaining curves ; hence 

 this case is reduced to the preceding; but it is necessary also, that the times of the 

 bodies in the two cases should be the same, in order that the places may cor- 

 respond, and consequently = , where v denotes the velocity of the body at 



any point of the curve b, from which equation can be deduced the correspondent 

 abscissae and ordinates, &c. of the curves b and a ; and thence the two cases are 

 reduced to the preceding, whence the correspondent forces in the directions of 

 the tangent, and perpendicular to it, can be found as above. 4. If some {m) 

 of the centres move in curves l, l', t,", &c. to be deduced from the laws of the 

 forces being given which act on them ; assume z and w, z' and u\ z", and u'\ &c. for 

 their respective abscissae and correspondent ordinates; and from them and y and 

 a:, y and .v, find the forces acting on the body moving in the curve required in the 

 direction of the tangent, and perpendicular to it, as before; then add all the 

 forces deduced which act perpendicular to the tangent, and also all contained in 

 the direction of the tangent together with the resisting force in the same direc- 

 tion, and let the sums resulting be respectively p and f': by the same method 

 find the sum of the forces which act on the bodies moving in l, l', l,", he, in 

 the directions of the tangents, and perpendiculars to them, which suppose s and 



