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

284 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 reolve it elf 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 intead of V we write the invere ratio of the centripetal force, and intead of PE the mean proportional between PS and 2LD; tho three parts will become ordinates to o many curve lines, whoe areas are dicovered by the common methods. Q. E. D.

If the centripetal force tending to the everal particles of the phere be reciprocally as the ditance; intead of V write PE the ditance; then $$\scriptstyle SL - \frac 12LD - \frac{ALB}{2LD}$$. Suppoe 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 decribe the rectangular area 2SL x AB; and the indefinite part LD, drawn perpendicularly into the ame length with a continued motion, in uch fort as in its motion one way or another it may either by increaing or decreaing remain always equal to the length LD, will derive $$\textstyle \frac {LB^2 - LA^2}{2}$$ 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 ame manner