Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/401

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Sect. XIII.
of Natural Philopoſophy.

the direction of lines parallel to that plane. And on the contrary if there be required the law of the attraction tending towards the plane in perpendicular directions, by which the body may be cauſed to move in any given curve line, the problem will be ſolved by working after the manner of the third problem.

But the operations may be contracted by reſolving the ordinates into converging ſeries. As if to a baſe A the length B be ordinately applied in any given angle, that length be as any power of the baſe \scriptstyle A^{\frac mn}; and there be sought the force with which a body, either attracted towards the baſe or driven from it in the direction of that ordinate, may be cauſed to move in the curve line which that ordinate always deſcribes with its ſuperior extremity; I ſuppoſe the baſe to be increaſed by a very ſmall part O, and I reſolve the ordinate \scriptstyle \overline {A+O} \vert^{\frac mn} into an infinite ſeries \scriptstyle A^{\frac mn} + \frac mnOA^{\frac {m - n}{m}} + \frac {mm - mn}{2nn}OOA^{\frac {m - 2n}{n}} &c. and I ſuppoſe the force proportional to the term of this ſeries in which O is of two dimenſions, that is to the term \scriptstyle \frac {mm - mn}{2nn}OOA \frac {m - 2n}{n}. Therefore the force ſought is as \scriptstyle \frac {mm - mn}{2nn}A \frac {m - 2n}{n}, or, which is the ſame thing, as \scriptstyle \frac {mm - mn}{2nn}B \frac {m - 2n}{n}. As if the ordinate deſcribe a parabola, m begin = 2, and n = 1, the force will be as the given quantity \scriptstyle 2B^0, and therefore is gi-.