398 PHILOSOPHICAL TRANSACTIONS. [aNNO 17 88. 



Exam. 2. Let the accelerating force be as the arc, that is, the distance from 

 the lowest point, and the resistance as the velocity; then will the fluxional equa- 

 tion (f — v) k = — vv be (ax — v) .v = — vv, which is an homogeneous equa- 

 tion of the first order: write in it zx for v, and its fluxion for v, and there 

 results the equation (ax — zx) X ^ = — s'ar^ir — zVr, whence (a — z) i — 



— zxz — z\v, and - = — ^ ^^ - „ and thence log. x= — ^ log. (a — z -|- z^) 



(w) — X cir. arc, whose radius is -|-i/(4a— l) and tangent (z — -i.) -j- b ; 



whence can be found v = xz, and from curvilinear areas i = -. 



V 



If 4a be less than 1 ; then it becomes log. x =: w — 



I , X log. " ~ ^, - «) " f -Lb ; where B is an invariable quantity to be 



assumed according to the conditions of the problem. 



Corol. If the force be directly as the distance, or as the arc of the curve 

 from the body to the lowest point, and the resistance as the velocity; then will 

 the velocity in one arc be to the velocity in the corresponding point of another 

 avcj as the arcs to be described; and consequently the times equal. 



4. If the body is acted on by forces tending to points s, s', s"', &c. situated 

 in different planes ; then let f be the sum of the forces in the direction of the 

 tangent at the point p ; f' and f'' the sum of the forces acting on the body in 2 

 different directions at the same point, which are not situated in the same plane 

 with the tangent and each other; from the 3 equations (f -f- x'v) a = — vv and 



— = 4- f' and — = 4-p'', in which the same letters denote the same quantities as 

 before, and c and C denotes the chords of curvature in the same directions as 

 the forces f' and f'', which from the curve being given can be found at any point ; 

 and if f' or f'' is given in terms of the distance from a given point, or an absciss 

 or ordinate, &c. the velocity v can be found in terms of the same, and x'v by 

 a simple algebraical equation : if f' is not given, and v is a given function of f , 

 substitute in v for v its value -/ (-i-c X f'), and there results an equation ex- 

 pressing the relation between f (which can be deduced from f' or p") and the 

 distance of the body from some given point, or the abscissae and ordinates of the 

 curve required, and their fluxions. If some of the forces act in parallel direc- 

 tions; the forces, velocities, &c. may be found by the same method. 



Prop. 10. — Fig. 13. Let a body be projected in a direction hl with a given 

 velocity, and be acted on by a force in a direction parallel to ap = x, which 

 varies as x a function of x; and also by another force in a direction parallel to 

 MP = y, that is perpendicular to ap, which force varies as y a function of y > 

 and let it move in a medium, of which the resistance is proportional to the velo- 

 city; to find the curve described. Find the fluent of (x -j- av) i= — vv, which 

 •corrected according to the conditions of the problem (viz. so that v at the 



