Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/397

Rh (by cor. 3. prop. 90.) the force with which any plane mHM attracts the point C, is reciprocally as $$\scriptstyle CH^{n - 2}$$. In the plane mHM take the length HM reciprocally proportional to $$\scriptstyle CH^{n - 2}$$, and that force will be as HM. In like manner in the everal planes IGL, nIN, oKO, &c. take the lengths GL, IN, KO, &c. reciprocally proportional to $$\scriptstyle CG^{n - 2}$$, $$\scriptstyle CI^{n - 2}$$, $$\scriptstyle CK^{n - 2}$$, &c. and the forces of thoe planes will be as the lengths o taken, and therefore the um of the forces as the um of the lengths, that is, the force of the whole olid as the area GLOK produced infinitely towards OK. But that area (by the known methods of quadratures) is reciprocally as $$\scriptstyle Cg^{n - 3}$$, and therefore the force of the whole olid is reciprocally as $$\scriptstyle CG^{n - 3}$$. Q. E. D.

Let the corpucle (Fig. 7.) be now placed on that hand of the plane IGL that is within the olid, and take the ditance CK equal to the ditance CG. And the part of the olid LGI x KO terminated by the parallel planes IGL, oKO, will attract the corpucle, ituate in the middle, neither one way nor another, the contrary actions of the oppoite points detroying one another by reaon of their equality. Therefore the corpucle C is attracted by the force only of the olid ituate beyond the plane OK. But this force (by cae 1.) is reciprocally as $$\scriptstyle CH^{n - 3}$$, that is (becaue CG, CK are equal) reciprocally as $$\scriptstyle CG^{n - 3}$$. Q. E. D.

Hence if the olid LGIN be terminated on each ide by two infinite parallel planes LG, IN; its attractive force is known, ubducting from the attractive force of the whole infinite olid LGKO, the attractive force of the more ditant part NIKO infinitely produced towards KO.