Page:Newton's Principia (1846).djvu/290

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284
[Book II.
the mathematical principles

II of this Book) the moment KL of AK will be equal to \scriptstyle \frac{2APQ+2BA\times PQ}{Z} or \scriptstyle \frac{2BPQ}{Z}, and the moment KLON of the area AbNK will be equal to \scriptstyle \frac{2BPQ\times LO}{Z} or \scriptstyle \frac{BPQ\times BD^{3}}{2Z\times CK\times AB}.

Case 1. Now if the body ascends, and the gravity be as AB² + BD², BET being a circle, the line AC, which is proportional to the gravity, will be \scriptstyle \frac{AB^{2}+BD^{2}}{Z}, and DP² or AP² + 2BAP + AB² + BD² will be AK \scriptstyle \times Z + AC \scriptstyle \times Z or CK \scriptstyle \times Z; and therefore the area DTV will be to the area DPQ as DT² or DB² to CK \scriptstyle \times Z.

Case 2. If the body ascends, and the gravity be as AB² - BD², the line AC will be \scriptstyle \frac{AB^{2}+BD^{2}}{Z}, and DT² will be to DP² as DF² or DB² to BP² - BD² or AP² + 2BAP + AB² - BD², that is, to AK \scriptstyle \times Z +

Principia1846-283.png


AC \scriptstyle \times Z or CK \scriptstyle \times Z. And therefore the area DTV will be to the area DPQ as DB² to CK \scriptstyle \times Z.

Case 3. And by the same reasoning, if the body descends, and therefore the gravity is as BD² - AB², and the line AC becomes equal to \scriptstyle \frac{BD^{2}-AB^{2}}{Z}; the area DTV will be to the area DPQ, as DB² to CK \scriptstyle \times Z: as above.

Since, therefore, these areas are always in this ratio, if for the area