Page:Newton's Principia (1846).djvu/248

 the attraction of that nearer part will, as the distance increases, decrease nearly in the ratio of the power CGn-3.

. 3. And hence if any finite body, plane on one side, attract a corpuscle situate over against the middle of that plane, and the distance between the corpuscle and the plane compared with the dimensions of the attracting body be extremely small; and the attracting body consist of homogeneous particles, whose attractive forces decrease in the ratio of any power of the distances greater than the quadruplicate; the attractive force of the whole body will decrease very nearly in the ratio of a power whose side is that very small distance, and the index less by 3 than the index of the former power. This assertion does not hold good, however, of a body consisting of particles whose attractive forces decrease in the ratio of the triplicate power of the distances; because, in that case, the attraction of the remoter part of the infinite body in the second Corollary is always infinitely greater than the attraction of the nearer part.

If a body is attracted perpendicularly towards a given plane, and from the law of attraction given, the motion of the body be required; the Problem will be solved by seeking (by Prop. XXXIX) the motion of the body descending in a right line towards that plane, and (by Cor. 2, of the Laws) compounding that motion with an uniform motion performed in 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, and that length be as any power of the base A$$\scriptstyle \frac{m}{n}$$; 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 $$\scriptstyle \overline{A+O}|\frac{m}{n}$$ into an infinite series $$\scriptstyle A\frac{m}{n}+\frac{m}{n}OA\frac{m-n}{n}+\frac{mm-mn}{2nn}OOA\frac{m-2n}{n}$$ &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