Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/378

290 and (by what Archimedes has demontrated) that uperficies will he as PF x DF x O. Let us uppoe beides the attractive forces of the particles of the phere to be reciprocally as that power of the ditances, of which n is index; and the force with which the uperhcies EFG attracts the body P, will be (by prop. 79.) as, $$\textstyle \frac {DE^2 \times O}{PF^n}$$ that is, as $$\textstyle \frac {2DF \times O}{PF^{n - 1}} - \frac {DF^2 \times O}{PF^n}$$. Let the perpendicular FN, drawn into O be proportional to this quantity; and the curvilinear area BDI, which the ordinate FN, drawn through the length DB with a continued motion will decribe, will be as the whole force with which the whole egment RBSD attracts the body P. Q. E. I.

To find the force with which a corpucle, placed without the centre of a phere in the axis of any egment, is attracted, by that egment.

Let the body P placed in the axis ADB of the egment EBK (Pl. 23. Fig. 6.) be attracted by that egment. About the centre P with the interval PE let the phærical uperficies EFK be decribed; and let it divide the egment into two parts EBKFE and EFKDE. Find the force of the firt of thoe parts by prop. 81. and the force of the latter part by prop. 83. and the um of the forces will be the force Pf the whole egment EBKDE. Q. E. I.