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

Rh 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 caued to move in any given curve line, the problem will be olved by working after the manner of the third problem.

But the operations may be contracted by reolving the ordinates into converging eries. As if to a bae A the length B be ordinately applied in any given angle, that length be as any power of the bae $$\scriptstyle A^{\frac mn}$$; and there be sought the force with which a body, either attracted towards the bae or driven from it in the direction of that ordinate, may be caued to move in the curve line which that ordinate always decribes with its uperior extremity; I uppoe the bae to be increaed by a very mall part O, and I reolve the ordinate $$\scriptstyle \overline {A+O} \vert^{\frac mn}$$ into an infinite eries $$\scriptstyle A^{\frac mn} + \frac mnOA^{\frac {m - n}{m}} + \frac {mm - mn}{2nn}OOA^{\frac {m - 2n}{n}}$$ &c. and I uppoe the force proportional to the term of this eries in which O is of two dimenions, that is to the term $$\scriptstyle \frac {mm - mn}{2nn}OOA \frac {m - 2n}{n}$$. Therefore the force ought is as $$\scriptstyle \frac {mm - mn}{2nn}A \frac {m - 2n}{n}$$, or, which is the ame thing, as $$\scriptstyle \frac {mm - mn}{2nn}B \frac {m - 2n}{n}$$. As if the ordinate decribe a parabola, m begin = 2, and n = 1, the force will be as the given quantity $$\scriptstyle 2B^0$$, and therefore is.