VOL. LXXVIII.] PHILOSOPHICAL TRANSACTIONS. 401 



the direction of the force (p) = -/(l — c*^) X 2 . ■■_^...'' ; and v the velocity of 

 the body in the curve, whose absciss is x and ordinate y. 



V ■'• " y 21 V "I 7/ 



In the same manner let -^=- x ,,. . .,> H- — =^ X 



sine of the angle made between the distance mm' and arc of the curve of which 

 the absciss is z and ordinate u, and consequently c' X p' will be the force in the 

 direction of its tangent, and therefore — vV = c' X p' X v^(i^ + «*) (a) and 

 v'^ = 4-cV), where c' is the chord of curvature in the direction of the force (p') 

 = V'(l — c'^) X 2 .^7-_— 4-, and v' the velocity of the body in the curve 

 whose absciss is z and ordinate u; then, because the times of describing cor- 

 respondent arcs in the two curves are equal, their increments will be equal, and 



consequently i = — - — ^ = — 5^, -!ii ; and there are deduced 5 fluxional equa- 

 tions, containing 6 variable quantities V, v', x, i/, z, and z^, and their fluxions; 

 reduce these equations, so that 4 of them (v, v, &c.) may be exterminated, and 

 there will result an equation expressing the relation between x and y the absciss 

 and ordinate of one curve, or z and u the absciss and ordinate of the other 

 curve, and their fluxions ; the fluential equation of which being found, and 

 properly corrected, gives the equation to the curve. The 5 equations are easily 

 reduced to 3 by exterminating the quantities v and v. The fluxional equation 

 resulting will most commonly be of the 5th order, as evidently appears from the 

 nature of the problem. 



2. The same principles may be applied to determine the curves, when the 

 bodies move in mediums, of which the resistances are given: for example, sup- 

 pose the resistances to vary as a function of the distance from a given point into 

 a function of the velocity: to the forces in the directions of the tangents con- 

 tained in the preceding case must be added or subtracted the given resistances 

 for the forces in the directions of the tangents, and the remaining process will 

 be the same as is before given. If two bodies describe similar orbits round a 

 common centre, either quiescent or moving uniformly in a right line; the forces 

 and velocities and resistances of the medium will be to each other in correspon- 

 dent points as the respective distances from the centre. 



Ppop. 13. — Given the forces acting on any bodies, and tending to points 

 either moveable or quiescent, or in the direction of the tangents, &c.; to find 

 the curve described by one of the bodies. 



1 . Assume a: and y for the absciss and ordinate of the curve required, and 

 from thence may be deduced the distances from any quiescent centre of force, and 

 consequently the force / in that direction; resolve it into 2 others, one in the 

 direction of the tangent, and the other in a different one ; for example, let it be in a 



VOL. XVI, 3 F 



