Page:Somerville Mechanism of the heavens.djvu/553

Chap II.] Chap. II.] MOTION OF THE MOON'S PBBIOBB. ^l

but the variation of the eccentricity, though small, has in the course of time a very great influence on the lunar motions. The mean motion of the moon is accelerated, when the eccentricity of the earth's orbit diminishes, which it has continued to do from the moat ancient observations down to our times ; and it will continue to be accelerated until the eccentricity begins to increase, when it will be retarded. In the interval between 1750 and 1850, the square of the eccentricity of the terrestrial orbit has diminished by 0.00000 140595. The corresponding increment in the angular velocity of the moon is the 0.0000000117821th part of this velocity. As this increment takes place gradually and proportionally to the time, its effect on the motion of the moon is less by one half than if it had been uniformly the same in the whole course of the century as at the end of it. In order, therefore, to determine the secular equation of the moon at the end of a century estimated from 1801, we must multiply the secular motion of the moon by half the very small increment of the angular velocity ; but in a century the motion of the moon is 1732559351". 544, which gives 10". 2065508 for her secular equation. Assuming that for 2000 years before and after the epoch 1750, the square of the eccentricity of the earth's orbit diminishes as the tone, the secular equa- tion of the mean motion will increase as the square of the time : it is sufficient then during that period to multiply 10'. 2065508 by the square of the number of centuries elapsed between the time for which we compute and the beginning of the nineteenth century ; hot in com- puting back to the time of the Chaldeans, it is necessary to carry the approximation to the cube of the time. The numerical formula for the acceleration is easily found, for since

2/3 m |«/(e~ - )dv

is the acceleration in the mean longitude of the moon, the true lon- gitude of the moon in functions of her mean longitude will contain the term

- 2/3 m f (e - e)ndt, I being the eccentricity of the terrestrial orbit at the epoch 1750. If then, I be any number of Julian years from 1750, by article 480, 2e' = 2e - 0". 171798t - 0".000068194f