VOL. XLII.] »1 PHILOSOPHICAL TRANSACTIONS. 640 



acceleration of a body that descends by its gravity in a medium which resists in 

 the duplicate ratio of the velocity. For when the utmost velocities, or limits, 

 are equal in those two cases, the time in which the issuing water acquires any 

 less velocity, is to the time in which the descending body acquires the same 

 velocity, as the area of the orifice to the area of the base; and if a cylindric 

 column be supposed to be erected on the orifice equal to the quantity of water 

 that issues at the orifice in the former of those times, the height of this column 

 will be to the space described by the descending body in the latter time, in the 

 same ratio as the orifice to the area of the base. The ratio of the force that 

 acts on the bottom of the vessel to the force that generates the motion of the 

 water issuing at the orifice, is deduced from Sir Isaac Newton's cataract, and is 

 the same that follows from the principle concerning the equality of the ascent 

 and descent of the centre of gravity, which was first applied to this inquiry by 

 Mr. Daniel Bernouilli, comment. Acad. Petrop. torn. 1. But there are several 

 precautions to be taken in applying this doctrine. 



After some other theorems concerning the centre of gravity, and several ob- 

 servations concerning the curvature of lines, and the angles of contact; the 

 author represents 4 general propositions in one view, that the analogy between 

 them may appear. The first gives the property of the trajectories that are de- 

 scribed by any centripetal forces, how variable soever these forces, or their 

 directions, may be. The second gives a like general property of the lines of 

 swiftest descent. The third gives the property of the line that is described in 

 less time than any other of an equal perimeter. And the fourth gives the pro- 

 perty of the figure that is assumed by a flexible line or chain, in consequence 

 of any such forces acting upon it. If we suppose k body to set out from any 

 point in the trajectory, or in the line of swiftest descent, with the velocity which 

 it has acquired there, and to move in the right line which is the direction of 

 the gravity, that results from the composition of the centripetal forces, then 

 shall its velocity, and its distance from the point where the perpendicular from 

 the centre of curvature meets that right line, flow proportionally, i. e. the 

 fluxion of the velocity, or of the right line that measures it, shall be to the 

 velocity, as the fluxion of that distance is to the distance. When the velocity 

 and direction of the motion is the same in the line of swiftest descent as in the 

 trajectory, their curvature is the same. Thus in the common hypothesis of 

 gravity, the curvature in the cycloid, the line of swiftest descent, is the same 

 as the parabola described by a projectile, if the velocities in those lines be equal, 

 and their tangents be equally inclined to the horizon. In order to find the na- 

 ture of the catenaria in any hypothesis of gravity, suppose the gravity to be in- 

 creased or diminished in the same proportion as the thickness of the chain 



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