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

Rh $$\scriptstyle I - \frac {PF}{PR}$$. The part I of this quantity, drawn into the length AB, decribes the area I x AB; and the other part $$\scriptstyle \frac {PF}{PR}$$ drawn into the length PB, decribes the area I into $$\scriptstyle \overline {PE - AD}$$ (as may be easily hewn from the quadrature of the curve LKI); and in like manner, the ame part drawn into the length PA decribes the area I into $$\scriptstyle \overline {PD - AD}$$, and drawn into AB, the difference of PB and PA decribes I into $$\scriptstyle \overline {PE - PD}$$, the difference of the areas. From the firt content I x AB take away the lat 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 $$\scriptstyle {AB - PB + PD}$$.

Hence alo is known the force by which a pheroid AGBC (Pl. 24. Fig. 4.) attracts any body P ituate externally in its axis AB. Let NKPM be a conic ection whoe 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 pheroid. From the vertices A, B, of the pheroid, let there be erected to its axis AB the perpendiculars AK, BM, repectively equal to AP, BP, and therefore meeting the conic ection in K and M; and join KM cutting off from it the egment KMRK Let S be the centre of the pheroid, and SC its greatet emi-diameter; and the force with which the pheroid attracts the body P, will be to the force with which a phere decribed with the diameter AB attracts the ame body,