Page:Newton's Principia (1846).djvu/448

10‴ 33^iv. as the aggregate of all the areas HpMh, generated in the time of one revolution of the point p (with due regard in summing to their proper signs + -), multiplied into AZ $\scriptstyle \times$ TZ $\scriptstyle \times {\frac {Pp}{PG}}$ to Mp $\scriptstyle \times$ AT³; that is, as the whole circle QAqa multiplied into AZ $\scriptstyle \times$ TZ $\scriptstyle \times {\frac {Pp}{PG}}$ to Mp $\scriptstyle \times$ AT³, that is, as the circumference QAqa multiplied into AZ $\scriptstyle \times$ TZ $\scriptstyle \times {\frac {Pp}{PG}}$ to 2Mp $\scriptstyle \times$ AT².

Cor. 3. And, therefore, in a given position of the nodes, the mean horary variation, from which, if uniformly continued through the whole month, that menstrual variation might be generated, is to 33″ 10‴ 33^iv. as AZ $\scriptstyle \times$ TZ $\scriptstyle \times {\frac {Pp}{PG}}$ to 2AT², or as Pp $\scriptstyle \times {\frac {AZ\times TZ}{{\frac {1}{2}}AT}}$ to PG $\scriptstyle \times$ 4AT; that is (because Pp is to PG as the sine of the aforesaid inclination to the radius, and $\scriptstyle {\frac {AZ\times TZ}{{\frac {1}{2}}AT}}$ to 4AT as the sine of double the angle ATn to four times the radius), as the sine of the same inclination multiplied into the sine of double the distance of the nodes from the sun to four times the square of the radius.

Cor. 4. Seeing the horary variation of the inclination, when the nodes are in the quadratures, is (by this Prop.) to the angle 33″ 10‴ 33^iv. as IT $\scriptstyle \times$ AZ $\scriptstyle \times$ TG $\scriptstyle \times {\frac {Pp}{PG}}$ to AT³, that is, as $\scriptstyle {\frac {IT\times TG}{{\frac {1}{2}}AT}}\times {\frac {Pp}{PG}}$ to 2AT, that is, as the sine of double the distance of the moon from the quadratures multiplied into $\scriptstyle {\frac {Pp}{PG}}$ to twice the radius, the sum of all the horary variations during the time that the moon, in this situation of the nodes, passes from the quadrature to the syzygy (that is, in the space of 177¹⁄₆ hours) will be to the sum of as many angles 33″ 10‴ 33^iv. or 5878″, as the sum of all the sines of double the distance of the moon from the quadratures multiplied into $\scriptstyle {\frac {Pp}{PG}}$ to the sum of as many diameters; that is, as the diameter multiplied into $\scriptstyle {\frac {Pp}{PG}}$ to the circumference; that is, if the inclination be 5° 1′, as 7 $\scriptstyle \times$ ⁸⁷⁴⁄₁₀₀₀₀ to 22, or as 278 to 10000. And, therefore, the whole variation, composed out of the sum of all the horary variations in the aforesaid time, is 163″, or 2′ 43″.