Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/374

Plate 23, Figure 3

L, A, S3, B, (Fig. 3.) erect the perpendicualrs Ll, Aa, Ss, Bb, of which ſuppoſe Ss equal to SI; and through the point s, to the aſymptotes Ll, LB, deſcribe the hyperbola asb meeting the perpendiculars Aa, Bb, in a and b; and the rectangle 2ASI ſubducted from the hyperbola AasbB, will leave ANB the area ſought.

Example 3. If the centripetal force tending to the ſeveral particles of the ſpheres decreaſe in a quadruplicate ratio of the diſtance from the particles; write $\scriptstyle {\frac {PE^{4}}{2AS^{3}}}$ for V, then $\scriptstyle {\overline {\sqrt {2PS+LD}}}$ for PE, and DN will become as $\scriptstyle {\frac {SI^{2}\times SL}{\sqrt {2SI}}}\times {\frac {1}{\sqrt {LD^{2}}}}-{\frac {SI^{2}}{2{\sqrt {2SI}}}}\times {\frac {1}{\sqrt {LD}}}-{\frac {SI^{2}\times ALB}{2{\sqrt {2SI}}}}\times {\frac {1}{\sqrt {LD^{2}}}}$ Theſe three parts drawn into the length AB, produce ſo many areas, viz. $\scriptstyle {2SI^{2}\times SL}{\sqrt {2SI}}$ into $\scriptstyle {\overline {{\frac {1}{\sqrt {LA}}}-{\frac {1}{\sqrt {LB}}}}}$; $\scriptstyle {\frac {SI^{2}}{\sqrt {2SI}}}$ into $\scriptstyle {\overline {\sqrt {LB-LA}}}$; and $\scriptstyle {\frac {SI^{2}\times ALB}{3{\sqrt {2SI}}}}$ into $\scriptstyle {\overline {{\frac {1}{\sqrt {LA^{2}}}}-{\frac {1}{\sqrt {LB^{2}}}}}}$. And theſe after due reduction come forth $\scriptstyle {\frac {2SI^{2}\times SL}{LI}}$, $\scriptstyle SI^{2}$ and $\scriptstyle SI^{2}+{\frac {2SI^{2}}{3LI}}$. And theſe by ſubducting the laſt from the firſt become $\scriptstyle {\frac {4SI^{2}}{3LI}}$. Therefore the entire force with which the corpuſcle P is attracted towards the centre of the ſphere is as $\scriptstyle {\frac {SI^{2}}{PI}}$, that us reciprocally as $\scriptstyle PS^{2}\times PI$. Q. E. I.