Page:Newton's Principia (1846).djvu/245

 the area 1 $$\scriptstyle \times$$ AB; and the other part $$\scriptstyle \frac{PF}{PR}$$, drawn into the length PB describes the area 1 into $$\scriptstyle \overline{PE-AD}$$ (as may be easily shewn from the quadrature of the curve LKI); and, in like manner, the same part drawn into the length PA describes the area 1 into $$\scriptstyle \overline{PD-AD}$$, and drawn into AB, the difference of PB and PA, describes 1 into $$\scriptstyle \overline{PE-AD}$$, the difference of the areas. From the first content 1 $$\scriptstyle \times$$ AB take away the last content 1 into $$\scriptstyle \overline{PE-AD}$$, and there will remain the area LABI equal to 1 into $$\scriptstyle \overline{PE-AD+PD}$$. Therefore the force, being proportional to this area, is as AB - PE + PD.

. 2. Hence also is known the force by which a spheroid AGBC attracts any body P situate externally in its axis AB. Let NKRM be a conic section whose ordinate ER perpendicular to PE may be always equal to the length of the line PD, continually drawn to the point D in which that ordinate cuts the spheroid. From the vertices A, B, of the spheriod, let there be erected to its axis AB the perpendiculars AK, BM, respectively equal to AP, BP, and therefore meeting the conic section in K and M; and join KM cutting off from it the segment KMRK. Let S be the centre of the spheroid, and SC its greatest semi-diameter; and the force with which the spheroid attracts the body P will be to the force with which a sphere described with the diameter AB attracts the same body 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 same principles may be found the forces of the segments of the spheroid.

. 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 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