Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/393

$\scriptstyle I-{\frac {PF}{PR}}$. The part I of this quantity, drawn into the length AB, deſcribes the area I x AB; and the other part $\scriptstyle {\frac {PF}{PR}}$ drawn into the length PB, deſcribes the area I 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 I into $\scriptstyle {\overline {PD-AD}}$, and drawn into AB, the difference of PB and PA deſcribes I into $\scriptstyle {\overline {PE-PD}}$, the difference of the areas. From the firſt content I x AB take away the laſt content I into $\scriptstyle {\overline {PE-PD}}$, and there will remain the area LABI equal to I into $\scriptstyle {\overline {AB-PE+PD}}$. Therefore the force being proportional to this area, is as $\scriptstyle {AB-PB+PD}$.

Plate 24, Figure 4

Cor. 2. Hence alſo is known the force by which a ſpheroid AGBC (Pl. 24. Fig. 4.) attracts any body P ſituate externally in its axis AB. Let NKPM 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 ſpheroid, 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,