or, since M is negligible in comparison with m,
em = n 2 a 3 . With these substitutions, and making? =?i 2 -j- 2, we have finally
o- *a A —w V 2 J
If we identify the coefficients of cos in the two expressions for T, we have
), 3 emm a
whence, by substitution and by neglecting small quantities,
We have also for V the two expressions
3 emm r . -2-73- sm and - maG l sin A:;
whence by identifying the coefficients
where n^ denotes the rate at which A: alters.
Solving the two equations, we deduce the following expressions:
= _ 3w 7i%! + 2?t 3 . ff = 3w Ti 2 ^^ 2 4 2w_ 1 _+_3 2 ) 2 7i"i 3 -77 2 7l 1 2 ~ "7l 1 2 (?l 1 ^ n 2 )~
We can now calculate the value of /j and q^ numerically. For
i _ 9 - 9 365-256 - 27 "322 - . .. . - 2 ~ - 2 =1 8504 365-256 1 178-72 /,=
-0-007204 71
and also
from these we deduce
The results at which we have arrived may be thus summarily stated, using, however, a more accurate value of g 1 than that which is formed by this method:
If we suppose the orbit of the moon to coincide with the plane of the ecliptic, and if we neglect the ellipticity of the orbit of the earth around the sun and that of the moon around the earth; if n denotes the mean motion of the earth around the sun, and n the mean motion of the moon around the earth; and, finally, if n and a be connected by the equation
7i 2 3 = 0-9972
then we have for the motion of the moon the equations
r = af 1-0-007204 cos (2nt-2n f) " sin (2nt-2n't).
We thus see that the motion of the moon, on the hypothesis which we have assumed, is different from a uniform circular motion. The distance from the moon to the earth is sometimes 1-1 39th part greater or less than its mean value. And the longitude of the moon is sometimes 39 30" in advance of or behind what it would be on the supposition that the moon was moving uniformly.
When the distance is the greatest, we have cos (2nt 2717)= -1; whence n, ~,, 2nt-2nt = ir or 3?r;
and it follows that the distance of the moon from the earth is great est at quadratures. [1]When the distance is least, then
cos (2nt-2n't)= +1, whence 2nt-2n't = or 2-ir;
consequently the moon is nearest to the earth at syzygy. It thus appears that the orbit of the moon as modified by the disturbing influence of the variation resembles an oval of which the earth is the centre, and of which the minor axis is constantly in the line of syzygies.
The mean longitude of the moon and the true longitude coincide when
This condition is fulfilled both at syzygy and quadrature; conse quently the mean place of the moon and its true place coincidewhen the moon is either in syzygy or in quadrature. The true place of the moon is at its greatest distance in advance of the mean place when
sin (2nt- 2n't) = l.
This condition is fulfilled at the middle points of the first and third quadrants, while at the middle points of the second and fourth quadrants
sin (2nt - 2n't) = - 1;
and therefore the moon is behind its mean place in the second and fourth quadrants.
After new moon, the distance between the moon and the earth gradually increases, and the apparent velocity of the moon also in creases until, when the moon is three or four days old, it has advanced 39 beyond its mean place; the velocity then begins to diminish, though the distance goes on increasing, until at first quarter the distance has attained a maximum. After first quarter the distance diminishes, and the moon falls behind its mean place, the maximum distance of 39 behind the mean place being reached about 11 days after new moon. At full moon the distance has become a minimum, and the mean place and true place coincide. At 18 days the true place has again gained 39 on the mean place, but the distance increases, and at third quarter the distance is again a maximum, and the true and the mean place coincide. After passing third quarter the distance diminishes, and the true place falls behind the mean place, the difference attaining a maximum on the 26th day, after which the true place gains on the mean place,. with which it coincides at new moon, when also the distance is again a minimum.
Since the amount of this irregularity in longitude is so consider able, being in fact larger than the diameter of the moon itself, it is very appreciable even in comparatively coarse observations. It was discovered by observation by Tycho Brahe, by whom it was named the variation.
It would lead us too far to endeavour to trace out any of the other irregularities by which the motion of the moon is deranged. We have taken the variation merely as an illustration of one of the numerous corrections which the law of gravitation has explained. The accordance which subsists between the values of these corrections as computed by theory and as determined by observation affords the most conclusive evidence of the truth of the law of universal gravitation.(r. s. b.)
GRAVITY, Specific. See Hydrodynamics.
GRAY, the chief town of an arrondissement in the department of Haute-Saône, is situated on the declivity of a hill on the left bank of the Saône, 37 miles S.W. of Vesoul by rail. Its streets are narrow and steep, but it possesses broad and beautiful quays, and the Allée des Capucins is a fine promenade. The principal buildings are the old castle of the duke of Burgundy, the church in the style of the Renaissance, the communal college (with a library of 15,000 volumes and a natural history museum), the theatre, and the barracks. The town possesses very large flour-mills, and among the other industries are ship building, dyeing, tanning, haircloth-weaving, plaster-casting, and the manufacture of machinery, oils, and starch. There is also a considerable trade in iron, corn, provisions, vegetables, wine, and wood. The population in 1876 was 7305. Gray was founded in the 7th century. Its former defensive works were destroyed by Louis XIV. in 1688. During the Franco-Prussian war General von Werder concentrated his army corps in the town, and held it for a month, making it the point d'appui of movements towards Dijon and Langres, as well as towards Besançon.
- ↑ 1 When the approximation is carried sufficiently far it is found that the coefficient of the variation is 39′ 30″.
