400 PHILOSOPHICAL TRANSACTIONS. [aNNO 1788. 



the curves a, a', a'', a'", &c. at the same time, and in the same plane, and the 

 forces of all the bodies acting on s, except two, s' and s''; to find the forces of 

 the 2 bodies s' and s'^ on the body s. — This proposition may be resolved by the 

 method given in prop. 4, for to produce the same effect the same finite forces 

 will be requisite, whether the centres of forces rest or move in given curves. 



1. 2. If the bodies s, s', s'', &c. move in different planes, then all the forces 

 acting on the body, except 3, may be given, which may be acquired from the 

 method given in the same proposition. Hence it appears, that 2w forces may 

 be requisite to be found from the conditions of the problem to determine all the 

 bodies to move in their respective curves, when they are all situated in the same 

 plane, and that 3 X w forces may be requisite in different planes, &c. if the 

 force of one body (s') on another (s'') does not at all depend on the force of the 

 same body (s') on any other (s''') ; and if the same can be praedicated of the rest, 

 then n .n — 3 forces of the above-mentioned bodies in the same, or w . n — 4 

 forces in different planes may be assumed at will. 



3. If the velocities f , v', v", &c. at every point of the arcs a, a, a", &c. of 

 the {n) above-mentioned curves a, a", a!\ &c. be given in terms of their arcs, 

 abscissae, or ordinates, &c. and the places in which the bodies are situated at the 

 same time in the arcs b, b\ b'\ &c. of some other curves b, b', b", &c. find 

 the corresponding velocities v, v', v'', &c. at the same time of the bodies in 

 the curves b, b', b'', &c. ; then make - = -, = -^, &c, =: -, or which is equal 



h 



v' 



b 



V' V 



toit = — or = — = &c. From the fluents of the fluxional equations resulting 



properly corrected will be found the arcs a, of, a\ &c., described by the bodies 

 in the curves a, a', a", &c. in the same time as the correspondent arcs b, b\ b'\ 

 &c.; and thence, by the method given in the preceding case, may be deduced 

 the forces. 



The same principles may be applied to bodies moving in resisting mediums. 



Prop. 12. — Given the law of the forces of 2 bodies acting on each other, to 

 find the 2 curves by them described. — Fig. 14. Assume x and^ for the absciss 

 (ap) and ordinate (pm) of one curve, and z and u for the absciss (ap') and ordi- 

 nate (p'm') of the other; where the abscissae ap and ap' begin from the same 

 point a, and are situated in the same line; then will the distance (d = m'm) 

 between the bodies = ^z + jc^ -f- "/u + 2/^; let the forces of the body placed at 

 M on that at m', and of the body placed at m' on that at m vary as (p : (d) = p, 

 and (p : (d) = f'; and let Mp=.r and pm = i/ ; then will cosine of the angle wmm' 



to radius (l) be"^— ± -jj-.-^-r^ + ^-^=^ X -.v^tt^tt = ^5 ^"^ consequently 



the force in the direction of the tangent mm will be c X f, whence — vv = 

 c X F X V {v* +^^) (a) and v^ = -^cp, where c is the chord of curvature in 



