Page:Newton's Principia (1846).djvu/447



an hour, since the angle GTg (by Prop. XXX) is to the angle 33″ 10‴ 33iv. as IT $$\scriptstyle \times$$ PG $$\scriptstyle \times$$ AZ to AT³, the angle GPg (or the horary variation of the inclination) will be to the angle 33″ 10‴ 33iv. as IT $$\scriptstyle \times$$ AZ $$\scriptstyle \times$$ TG $$\scriptstyle \times$$ $$\scriptstyle \frac{Pp}{PG}$$ to AT³. Q.E.I.

And thus it would be if the moon was uniformly revolved in a circular orbit. But if the orbit is elliptical, the mean motion of the nodes will be diminished in proportion of the lesser axis to the greater, as we have shewn above; and the variation of the inclination will be also diminished in the same proportion.

. 1. Upon Nn erect the perpendicular TF, and let pM be the horary motion of the moon in the plane of the ecliptic; upon QT let fall the perpendiculars pK, Mk, and produce them till they meet TF in H and h; then IT will be to AT as Kk to Mp; and TG to Hp as TZ to AT; and, therefore, IT $$\scriptstyle \times$$ TG will be equal to $$\scriptstyle \frac{Kk\times Hp\times TZ}{Mp}$$, that is, equal to the area HpMh multiplied into the ratio $$\scriptstyle \frac{TZ}{Mp}$$: and therefore the horary variation of the inclination will be to 33″ 10‴ 33iv. as the area HpMh multiplied into $$\scriptstyle AZ\times\frac{TZ}{Mp}\times\frac{Pp}{PG}$$ to AT³.

. 2. And, therefore, if the earth and nodes were after every hour drawn back from their new and instantly restored to their old places, so as their situation might continue given for a whole periodic month together, the whole variation of the inclination during that month would be to 33″