Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/246

 170 is equal to the quare of DE. and therefore the accelerations generated in the paage of the bodies from D and I to F and K are equal. Therefore the velocities of the bodies in E and K are alo equal: and by the ame reaoning they will always be found equal in any ubequent equal ditances. Q. E. D.

By the ame reaoning, bodies of equal velocities and equal ditances from the centre will be equally retarded in their acent to equal ditances. Q. E. D.

Therefore if a body either ocillates by hanging to a tring, or by any polihed and perfectly mooth impediment is forced to move in a curve line; and another body acends or decends in a right line, and their velocities be equal at any one equal altitude; their velocities will be alo equal at all other equal altitudes. For, by the tring of the pendulous body, or by the impediment of a veel perfectly mooth, the ame thing will be effected, as by the tranvere force NT. The body is neither accelerated nor retarded by it, but only is obliged to quit its rectilinear coure.

Suppoe the quantity P to be the greatet ditance from the centre to which a body can acend, whether it be ocillating, or revolving in a trajectory, and o the ame projected upwards from any point of a trajectory with the velocity it has in that point. Let the quantity A be the ditance of the body from the centre in any other point of the orbit; and let the centripetal force be always as the power $$\scriptstyle{A^{n-1}}$$ of the quantity A, the index of which power n-1, is any number n diminihed by unity. Then the velocity in every altitude A will be as $$\scriptstyle{\sqrt{P^n-A^n}}$$, and therefore will be given. For by prop. 59. the velocity of a bo-