Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/394

as $\scriptstyle {\frac {AS\ timesCS^{2}-PS\times KMRK}{Ps^{2}+CS^{2}-AS^{2}}}$ is to $\scriptstyle {\frac {AS^{3}}{3PS^{2}}}$. And by a calculation founded on the ſame principles may be found the forces of the ſegments of the ſpheroid.

Plate 24, Figure 5

Cor. 3. If the corpuſcle be placed within the ſpheroid and in its axis, the attraction will be as its diſtance from the centre. This may be eaſily collected from the following reaſoning, whether the particle be in the axis or in any other given diameter. Let AGOF (Pl. 2.4. FQ. 5.) be an attracting ſpheroid, S its centre, and P the body attracted. Through the body P let there be drawn the ſemi-diameter SPA, and two right lines DE, FG meeting the ſpheroid in D and E, F and G; and let PCM, HLN be the ſuperficies of two interior ſpheroids ſimilar and concentrical to the exterior, the firſt of which paſſes through the body P, and cuts the right lines DE, FG in B and C; and the latter cuts the ſame right lines in H and I, K and L. Let the ſpheroids have all one common axis, and the parts of the right lines intercepted on both ſides DP and BE, FP and CG, DH and IE, FK and LG will be mutually equal; becauſe the right lines DE, PB, and HI are biſſected in the ſame points as are alſo the right lines FG, PC and KL. Conceive now DPF, EPG to repreſent oppoſite cones deſcribed with the infinitely ſmall vertical angles DPF, EPG, and the lines DH, EI to be infinitely ſmall alſo. Then the particles of the cones DHKF, GLIE, cut off by the ſpheroidical ſuperficies, by reaſon of the equality of the lines DH and EI, will be to one another as the ſquares of the diſtances