VOL. LXXVIII.] PHILOSOPHICAL TBANSACTIONS. 403 



s, s' and s% /', s'', &c: then reduce the '2 (w -f- equations of the formulae found 

 above, viz. v=.y X . .. ... and — w = f /(.r -|- y) ; v^ = s X -V ..^ --.■-- 



and - 2;'^' =s X /(i' + «'); t;^'" = ^ X ^.$^'?^* and - v<> X f'^ = s' X -/ {z"^ 

 + /^)} &c. where v, v", v"\ &c. respectively denote the correspondent veloci- 

 ties of the bodies moving in the curves whose abscissae are ar, z, z', z", &c. ; 



and also the (m + l) equations - = ■ ^, = -^— ^; — • = ^^^^^ — = 



'^^ ^^"^ ' = &c. containing the 3 (m + l). + 1 variable quantities x and y, z 

 and w, z' and u\ z" and m'', &c., v, v, v\ &c. and the variable quantity con- 

 tained in B and v, into 1, so that all the variable quantities except x and^ and 

 their fluxions may be exterminated, and there results an equation to the curve 

 required expressing the relation between x and y its absciss and ordinate, and 

 their fluxions. 5, If the forces are not situated in the same plane, assume x, 

 X and y, for the 2 abscissae and ordinates of the curve required; and z, z and u\ 

 z' and u\ z", z" and u"\ &c. for the 2 abscissae and ordinates of the (m) curves 

 L, l', i/'f &c. respectively; and from the preceding method may be acquired the 

 3 (m + l) equations ?;* = p X c, ?;* = f' X c', and — vv = f'' X \^{k + i^ 

 + y); i;'* = s X C, v"" = <TC and — vv' = s X >/(z* + i* + «*); t;'''* = s'c^ 

 = (tV and - v"v" = s'X -/(z'* + -s"' + t**); «^'"* = s''c'" = (rV" and - ^v" 

 = s" X \^ iz"^ + z^' + « ^ ) ; in which v denotes the velocity in the required 

 curve, andi;' v", v", &c. the correspondent velocities in the curves l, l', l" &c.; 

 and p, p', and f"; s, s-and^; s', tr' and /; s", /'and/'; &c. denote the forces 

 acting on the respective bodies in 2 diflerent planes and in the tangents, which 

 planes cut each other in the tangents of the curves ; and c and c, &c., c' and c , 

 &c., c" and c", &c., the 4 chords or radii of curvature in those 2 planes to the 

 different curves in the directions of the forces; and also the {m -{- \) equations 

 before-mentioned- = — ^ — = -, = — ^ ~,~^ — ^=&c. 



where /(x* + i' + y*), \/(z'* + -s'* + « *)> &c- are the fluxions of the arcs of 

 the required curve, and of the curves, l, l', l", &c. reduce these 4m -\- 4 equa- 

 tions containing 4m 4- 5 variable quantities into 2, so that all the variable quan- 

 tities except 3, x, x, and 3/, and their fluxions may be exterminated; and there 

 result the 2 equations required. 



It may be observed, that when the resistance, arising from the density of the 

 medium and velocity {v) of the body, varies as x' X t^* + x, where x and x' 

 are as functions of the distances from the given points, the resolu- 

 tion of the fluxional equations will generally be more easy, than when the 

 resistance varies as other functions of the velocities. If the force acts equally 

 on the particles of the body and fluid, then the force by which a body descends 

 in a medium is as the whole force x acting on the body at the given distance, 



3p2 



