VOL. LXXVIII.] PHILOSOPHICAL TRANSACTIONS. 3QQ 



point H may be to the velocity of projection :: hc : h^, where ^cis drawn perpendi- 

 cular to ap) suppose 2; = x'; find the fluent of —, which corrected so as to be- 

 come =: O, when X = ah, let be x". In the same manner find the fluent of 

 (y -{■ aV) ^ = — vvi which corrected, so that v' at the point h may be to the 



velocity of projection :: cb : hZ>, suppose i;' := y'; find the fluent of ~, which cor- 

 rected so as to become = O, when pm = O, let be = y"'; assume x'' = y'', and 

 thence from x find y : take ap = x and pm = z/, and m will be a point in the 

 curve, which a body projected in the line hl describes ; and if Mm in the direc- 

 tion parallel to hap : mo perpendicular to it :: velocity v : velocity v' then will mo 

 be a tangent to the curve in the point m. 



2. If a body is acted on by forces tending to any given points s, s', s", &c. 

 which vary as given functions of their distances from the body, and resisted by 

 a force which varies according to a given function v of the velocity (v) into its 

 density x', where x' varies according to some function of the distances from the 

 given points, &c. ; to find the curve described. 



1 . From the distances of the body from 2 given points, or the absciss and 

 ordinate of the curve described, and their fluxions, &c. find the forces acting in 

 the direction of the tangent to the curve, and in some other direction, which 

 suppose F and p'; and also the chord of curvature in the above-mentioned 

 direction, which let be c ; then from the equations (p -f- v X x') k = — vv and 

 t;'^ = -L c X F reduced into one by writing for v its value in the function v, and 

 for vv its value deduced from the equation i;^ = ^-c X f, and for A (the fluxion 

 of the arc) its value deduced from the distances, &c. will result an equation 

 expressing the relation between the distances from 2 given points to the curve, 

 or its absciss and ordinates, and their fluxions. 



3. If the forces are not all situated in the same plane, then from the before 

 given equation (f -f v X x') a = — rvy and the 2 others v"^ = ^c X f' and 

 t;* = 4-c'p'', where f denotes the force in the direction of the tangent, and p' and 

 p" are the forces in different directions, which both are not situated in the same 

 plane with each other and the tangent, and in which directions the chords of 

 curvature are respectively c and-c'; since the quantities f, f', and p"'; c and c' 

 and A (as proved before) can all be expressed in terms of the distances from 3 

 given points, or from 2 abscissae and 1 ordinate, and their respective fluxions; 

 may be deduced 2 fluxional equations expressing the relation between the dis- 

 tances from 3 given points, or 2 abscissae and an ordinate, &c. The same prin- 

 ciples may be applied to cases, in which some of the forces act in parallel 

 directions. 



On Moveable Centres. 

 Pbop. 11. — 1. Given the respective places of (n) bodies s, s', s"', s'^", &c. in 



