Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/270

 in a greater than a triplicate ratio of the altitude, the body at its parting from the apis, will either decend to the centre or acend in infinitum, according as it decends or acends at the beginning of its motion. But if the force in its reces from the centre either decreaes in a les than a triplicate ratio of the altitude, or increaes in any ratio of the altitude whatoever; the body will never decend to the centre, but will at ome time arrive at the lower apis; and on the contrary, if the body alternately acending and decending from one apis to another never comes to the centre, then either the force increaes in the reces from the centre, or it decreaes in a les than a triplicate ratio of the altitude; and the ooner the body returns from one apis to another, the farther is the ratio of the forces from the triplicate ratio. As if the body hould return to and from the upper apis by an alternate decent and acent in 8 revolutions, or in 4, or 2. or 1$$\scriptstyle \frac 12$$; that is if an hould be to n as 8 or 4 or or 2 or 1$$\scriptstyle \frac 12$$, and therefore $$\textstyle \frac {nn}{mm} - 3$$ be $$\scriptstyle \frac 1{64} - 3$$, or $$\scriptstyle \frac 1{16} - 3$$, or $$\scriptstyle \frac 14 - 3$$, or $$\scriptstyle \frac 49 - 3$$, then the force will be as $$\scriptstyle A ^{\frac 1{64} - 3}$$, or $$\scriptstyle A ^{\frac 1{16} - 3}$$, or $$\scriptstyle A ^{\frac 14 - 3}$$, or $$\scriptstyle A ^{\frac 49 - 3}$$; that is, it will be reciprocally as $$\textstyle A^{3 - \frac 1{64}}$$, or $$\textstyle A^{3 - \frac 1{6}}$$, or $$\textstyle A^{3 - \frac 14}$$, or $$\textstyle A^{3 - \frac 49}$$, If the body after each revolution returns to the ame apis, and the apis remains unmoved, then m will be to n as 1 to 1, and therefore $$\textstyle A ^{\frac {nn}{mm} - 3}$$ will be equal to $$\textstyle A^{-2}$$ or $$\textstyle \frac 1{AA}$$ and therefore the decreae of the forces will be in a duplicate ratio of the altitude; as was demontrateds above. If the body in three fourth parts