40 PHILOSOPHICAL TRANSACTIONS. [anNO 1756. 



they are as those errors or motions, of which they are the equations, generated 

 in those times : but the periodic times are very nearly as the equations. Hence, 

 because of the given errors of the lunar and terrestrial motions, and the pe- 

 riods of the equations of the lunar errors, by analogy those of the terrestrial 

 errors will be deduced. 



Thus, the period of the equation of the lunar apogee, and of the variation of 

 the equation of the moon's centre, since it is proportional to the sun's revolution 

 at the moon's apogee ; and because of the similitude of forces similarly applied, 

 the period of the equation of the earth's aphelion, and of the variation of the 

 equation of the centre, must be proportional to the revolution of Jupiter at the 

 earth's aphelion, therefore those lunar equations will be to these like equations 

 of the earth, as the motion of the lunar apogee, in the time of the sun's revo- 

 lution at the moon's apogee, is to the motion of the earth's aphelion in the time 

 of Jupiter's revolution at the earth's aphelion ; that is, the mean annual motion 

 of the moon's apogee being 40° 40' 43 ", and the annual motion of the earth's 

 aphelion above found ] 3" l"' 1^"", as 45° 5 1' 40' to 2' 3A" 42'". Therefore, 

 putting the total variation of the greatest equation of the moon's centre = 2° 

 41 '4^, as it is nearly in astronomical tables, the variation of the greatest equa- 

 tion of the centre of the earth or sun will be Q'^ 4"'. Now, let e denote the 

 greatest mean equation of the sun's centre, then will e + ^" 32'" be the gi-eatest 

 equation, and e — ^)l" yi'" the least equation; and by these equations there will 

 also be given the corresponding excentricities. 



Then, like as the variation of the greatest equation of the moon's centre in- 

 creases in the duplicate ratio of the sioe of the distance of the moon's apogee 

 from its quadrature with the sun, so the variation of the greatest equation of the 

 sun's centre, that is, the increment of the greatest equation, is augmented in 

 the duplicate ratio of the sine of the distance of the earth's aphelion from its 

 quadrature with Jupiter ; or, the variation of the mean equation is to half the 

 total variation, viz. to A" 32'", as the cosine of double the distance of Jupiter 

 from the earth's aphelion is to radius ; then adding this to the mean equation 

 when the line of the apses of the orbis magnus passes from its octant with Jupiter 

 to the syzygies, or from the syzygies to the octants ; in the other parts it is 

 subducted. 



In like manner, if the greatest equation of the moon's apogee be stated at 12** 

 18', then 45° 5 1' 40' will be to 2' 3 A" 42'", as 12° 18' is to the greatest equa- 

 tion of the motion of the earth's aphelion, or of the sun's apogee, which there- 

 fore will be 41 " 30", viz. where the apses of the earth's orbit is in its octant with 

 Jupiter. In other positions the equation of the aphelion will be to the greatest 

 equation, as the sine of double the distance of Jupiter from the earth's aphelion, 

 is to radius, adding it to the mean motion in the transit of the apses of the 



