Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/372

quantity $\scriptstyle {\frac {DE^{2}\times PS}{PE\times V}}$, which (by cor. 4. of the foregoing prop.) is as the length of the ordinate DN will now reſolve it ſelf into three parts $\textstyle {\frac {2SLD\times PS}{PE\times V}}-{\frac {LD^{2}\times PS}{PE\times V}}-{\frac {ALB\times PS}{PE\times V}}$; where if inſtead of V we write the inverſe ratio of the centripetal force, and inſtead of PE the mean proportional between PS and 2LD; thoſ three parts will become ordinates to ſo many curve lines, whoſe areas are diſcovered by the common methods. Q. E. D.

Example 1. If the centripetal force tending to the ſeveral particles of the ſphere be reciprocally as the diſtance; inſtead of V write PE the diſtance; then $\scriptstyle SL-{\frac {1}{2}}LD-{\frac {ALB}{2LD}}$. Suppoſe DN equal to its double $\scriptstyle 2SL-LD-{\frac {ALB}{LD}}$; and 2SL the given part of the ordinate drawn into the length AB will deſcribe the rectangular area 2SL x AB; and the indefinite part LD, drawn perpendicularly into the ſame length with a continued motion, in ſuch fort as in its motion one way or another it may either by increaſing or decreaſing remain always equal to the length LD, will deſrive $\textstyle {\frac {LB^{2}-LA^{2}}{2}}$ that is, the area SL x AB; which taken from the former area 2SL x AB leaves the area SL x AB. But the third part $\textstyle {\frac {ALB}{LD}}$, drawn after the ſame manner