VOL. LXXVIII.] PHILOSOPHICAL TRANSACTIONS. 3Q3 



proposition find the curve which a body, agitated by forces/ X 



— , /' X At-,/'' X ^, &c. tending to the points s, s', s", &c. describes, and it 



will be the curve required. 



4, If the body moves in a curve of double curvature, and the forces/,/',/", 

 &c. tending to all the centres m, m', m", m'", &c. be given; from the fluent of 



the fluxional quantity (/ x ^ ±/' X J^ ±/' X ^^ ±/" X ^^~ &c.) X. 

 (a denoting the same quantity as before) =/ X mp ±/ X m'p ± /" X m"p ±/"' 

 X m"'p ± &c. =/ X d ±/' X D ±/" X d'' ±/'"X d" ± &c. = B= — ,,^ (/ 

 /"',/",/", &c. being given functions of d, d', d", &c. respectively) can be deduced 

 the square of the velocity = — 2z, which will be a function of d, d', d", d'", 

 d"", &c., and consequently a function of d, d', d", easily to be derived: substitute 

 this function — 2z for v^ in the two following equations — , = f' and %, = v\ 

 where r' and r" denote the radii of curvature in two different planes of which 

 the tangent above-mentioned in prob. 4, art. 4, is their intersection, and f' and 

 p" the sum of the forces in lines perpendicular to the tangent, and in the res- 

 pective planes: from these forces, calculated in terms of the distances from 3 

 given points d, d', and d"; or in terms of 2 abscissae and 1 ordinate, and from 

 the radii r' and r" may be deduced 1 fluxional equations of the 2d order, ex- 

 pressing the relation between 3 distances d, d', and d", &c. which may always be 

 reduced to 1 fluxional equation of the 4th order, expressing the relation between 

 1 absciss and its correspondent ordinates, or the distances from 2 given points. 



5. The general fluxional equation expressing the relation between the distances 

 from 2 given points will be of the 4th order, if no fluents are contained in it; 

 for it admits of 4 different quantities to be assumed at will, or according to the 

 conditions of the problem. 



6. If some points, to which the forces tend, are situated at an infinite dis- 

 tance; that is, some forces always act parallel to themselves; from the given 

 forces acting either to given points, or in parallel directions, by the equation/ 

 X i) ± /' X i)' +/" X d" — &c. = — r, can be deduced the square of the velocity 

 at a point p in terms of the distances from 2 given points, or of an absciss and 

 ordinate; if the centres, &c. and parallel forces are all situated in the same 

 plane: or in terms of the distances from 3 points, or 2 abscissae and an ordinate, 

 if situate in different planes; from the centres, &c. and forces given, find the 

 sum p of the forces in any direction (pl) (the direction of the tangent excepted) 

 acting on the body at the point p, and the chord of curvature c of the curve 

 at the same point and in the same direction; in the equation v^ = -lf X c for 

 v^ substitute the value before found, and there results an equation expressing the 

 relation between the distances from 2 points, or an absciss and ordinate, &c. 

 if the forces act in the same plane: but if the forces act in different planes, 



VOL. XVI. 3 E 



