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

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

If the corpucle be placed within the pheroid and in its axis, the attraction will be as its ditance from the centre. This may be eaily collected from the following reaoning, whether the particle be in the axis or in any other given diameter. Let AGOF (Pl. 2.4. FQ. 5.) be an attracting pheroid, S its centre, and P the body attracted. Through the body P let there be drawn the emi-diameter SPA, and two right lines DE, FG meeting the pheroid in D and E, F and G; and let PCM, HLN be the uperficies of two interior pheroids imilar and concentrical to the exterior, the firt of which paes through the body P, and cuts the right lines DE, FG in B and C; and the latter cuts the ame right lines in H and I, K and L. Let the pheroids have all one common axis, and the parts of the right lines intercepted on both ides DP and BE, FP and CG, DH and IE, FK and LG will be mutually equal; becaue the right lines DE, PB, and HI are biected in the ame points as are alo the right lines FG, PC and KL. Conceive now DPF, EPG to repreent oppoite cones decribed with the infinitely mall vertical angles DPF, EPG, and the lines DH, EI to be infinitely mall alo. Then the particles of the cones DHKF, GLIE, cut off by the pheroidical uperficies, by reaon of the equality of the lines DH and EI, will be to one another as the quares of the ditances