39*2 PHILOSOPHICAL TRANSACTIONS. £aNNO 1788. 



titles: the same may be predicated of similar lines drawn to the centres s"', s"", 

 &c.; and consequently (/ X ^ ± / X ^^ ±/' X ^ ±/"' X ^' ± &c.) 

 X A (where a, as before, denotes the fluxion of the arc of the curve) = f x d 

 ±J' X d' ±/" X b" ±f"' X d" ± &c. = — vvi if V denotes the velocity; but 

 as/,/',/",/'", &c. are functions of d, d', d", d"', &c. respectively, t\\e fluent 

 of the above-mentioned quantity /d ±/d' ±f"T>" ± &c. can be found in terms 

 of D, d', d", d'", &c. from the fluents of the fluxions /b,/'i', &c.; and conse- 

 quently in terms of d and d', which let be z, then will z = 



=-^'; buti;^=-2z = poX(/X-±/X?-'±/"X 5^±/'" X~ ± 



2 ^-^ SP -^ SP "^ S F -^ S P 



&c.) a fluxional equation of the second order expressing the relation between d 

 and d', and their fluxions. 



2. To find an equation expressing the relation between a: = sm and 7/ = mp, 

 where sm {x) is the absciss beginning from s and continued in the line ss', and 

 MP (2/) the perpendicular ordinate of the curve described by a body acted on by 

 the above-mentioned forces ; in the fluxional equation found before for d and d' 

 and their fluxions substitute (.r^ -j- y^)^ and ((ss' ± xy -\- y'^y and their fluxions, 

 and there results the equation sought. 



Corol. It easily appears, that the general fluent may contain two invariable 

 quantities to be assumed at will, or according to the conditions of the problem ; 

 that is, at a given distance the velocity and the direction may be assumed at will, 

 and consequently the general fluxional equation, expressing the above-mentioned 

 relation, will be of the second order, if no fluents are contained in it. 



Corol. From vy and pz/', and the points s and s' being given, can easily be 

 deduced geometrically the direction of the tangent and the lines sy, sz/', &c. ; for 

 divide the line ss' in r, so that vy ± p/ : ss' :; vy : sr, and through r draw the 

 line pr, the perpendicular to vr through p will be the tangent t/p/; to this line 

 the perpendiculars from s and s' will be the lines sy and s'/ required. 



Corol. From the fluent of the above-mentioned fluxional equation may be 



deduced the velocity v in terms of d and d'; and from the fluent of ° ^ ° which 

 •^ p_y X V 



is a function of d multiplied into d, may be deduced the time. 



3. If the plane in which the body (p) moves, and all the forces /',/", /", &c. 

 tending to points m', m", m'", &c. not situated in the same plane (except one / 

 tending to a given point m) be given; then the force tending to that point can 

 be found, and the curve described. Resolve all the forces tending to the points 

 M, M', m", m'", &c. into two others; one ms, m's, m"s, m"'s, &c. perpendicular 

 to the plane in which the body moves, and the other sp, s'p, s"p, s"'p, &c. in the 



plane; then will / X — ± /' X —r- ±.f" X —77- ± &c. = 0, from which equa- 

 r ^ ^p "^mp-'mp ' ^ 



tion/the force tending to the point m may be found; then, from the preceding 



