Centripetal Forces, 99 
before given, be deduced the forces in the two above mentioned 
directions ; and in the fame manner may be found from x , y, 
„v, andy, the forces m the directions of the tangent and per- 
pendicular to it, which follow from all the forces tending to 
given points, and adting on the body moving in the curve to be 
inveftigated. 2. If fome of the centers of force move in 
given curves B, B', B", &c. whofe arcs let be denoted 
by B, B', &c. and their refpeCtive places at the fame time 
are given ; then from their rcfpeCtive places given and forces, 
and .v and y , and a* and y, can, as before, be deduced the 
forces in the direction of the tangent and its perpendicular to 
the curve required. 3. If other centers of forces move in 
given curves A, A', A", &c. and the velocities are given at 
every point of the curves ; let A, A ', A /x , &c. be the arcs of 
the curves A, A', A", Sec. and fuppofe v, v', v ", See. their 
correfpondent velocities ; then, if the increments of the time 
A 4/ a " 
be given, will — — — -r=&c. but as the velocities are given 
& V V v ° 
at every point of the curves, v in the curve (A) will be given 
in terms of its ablcifs, ordinate, arc, &c. and confequently 
— in terms of the fame quantities and their firft fluxions ; the 
fame may be affirmed of the fluxions 
in the curves A", 
V 
A ", &c. j hence, from the equation — — ^ , can be deduced 
the relation between the ablcifs or ordinate, See. of the curve 
A and its correfpondent abfcils or ordinate, Sec. of the curve 
A / ; and fo of the remaining curves ; hence this cafe is reduced 
to the preceding ; but it is neceflary alio, that the times of the 
bodies in the two cafes fhould be the fame, in order that the 
• • 
places may correfpond, and confequently — , where V 
O 2 denotes 
