earth in the periphery of the ellipse at any time, when the radius vector is represented by r; that is, if it represent an arc whose radius is unity, which is described by a point in SE in one second of time; then (art. 196.) r2v is the sectoral area described by SE in one second of time. But, by conic sections, CP being supposed to be unity, T√(1—e2) expresses the area of the ellipse; therefore, by the proportionality of the times to the areas (art. 194.),

and the last expression is the value of the radius vector at the time when the equation of the centre is the greatest. Now, in general (art. 204.), r = (1-e cos. u) and r = 1-e2

1+e cos. @; and, substituting the above value of r in each of these equations, we have

or

—e2)=1—e cos. u, also (1 —e2)*

1-e2

=1+e cos. 0 '

l—e2)*=1—e cos. u, and 1+e cos. 0= (1—e2)*.

Developing by the binomial theorem, and neglecting powers of e above the square, we have

whence

1-e2=1-e cos. u, also 1+e cos. 0=1-že2;

cos. u= e and cos. 0 -že.

But at the time when the equation of the centre is the greatest, u and are very nearly equal to right angles, u being greater; therefore, for u putting

being less, and

Then, since the equation of the centre is (art. 206.) equal to ent, or (art. 203. (A)) equal to

-ue sin. u;

M

substituting for 0 and u their values at the time when the equation of the centre is the greatest, and representing that greatest equation by E (expressed by a circular arc whose radius is unity), we have

From the equations (A) (B) and (c) (arts. 203, 204.) there may be obtained by the formulæ for the developments of functions in infinite series, values of 0, of r, and of 0-nt, in ascending powers of e, with sines and cosines of arcs which are multiples of nt, the mean anomaly.

208. Astronomers have computed the mean movement of the sun, or the angular space which the luminary would appear to describe about the earth in a day, an hour, &c. if its movement were quite uniform, or such as it might be supposed to have if it revolved in a circular orbit undisturbed by any attractions exercised upon it by the other bodies of the system; and they have added the mean place of the sun in the ecliptic, that is, its distance in longitude from the mean position of the equinoctial point, at a certain epoch, as the commencement of a particular year or century, together with its mean place in longitude at the commencement of the preceding and following years, and supplementary tables by which the use of the table may be extended to 10,000 years before and after the nineteenth century. They have also computed the mean longitude of the perigeum for the epoch, with the mean position of that point at the commencement of each year and the mean daily movement of the same point, in longitude; and, from the tables of such positions and movements, it is evident that the mean longitude of the sun and of the perigeum, or the distance of either from the mean place of the equinoctial point, can be computed for any given time by a simple addition of the numbers.

Again, the value of the equation of the centre corresponding to any given mean anomaly (mean distance of the earth from the perihelion point of its orbit) is also given in astronomical tables; and these values being applied to the mean longitude

above mentioned, there results the longitude of the sun for a perfectly elliptical orbit: the effects of the perturbations produced by the different planets are also computed, in order that, being applied to the elliptical place of the sun, the true longitude for the given instant may be obtained. Lastly, the values of the radius vector of the earth's orbit, computed on the supposition that the orbit is an ellipse and that the semiaxis major is equal to unity, are given, and to these are added tables of the variations produced in that element by planetary perturbations.

CHAP. IX.

THE ORBIT OF THE MOON.

THE FIGURE OF THE MOON'S ORBIT.—PERIODICAL TIMES OF HER REVOLUTIONS. THE PRINCIPAL INEQUALITIES OF HER MOTION. HER DISTANCE FROM THE EARTH EMPLOYED TO FIND APPROXIMATIVELY THE DISTANCE OF THE EARTH FROM THE SUN.

209. The processes employed to determine the apparent path of the moon in the celestial sphere correspond to those which have been explained in speaking of the sun. The right ascension and declination of the moon are to be observed every day when she comes to the meridian; and the obliquity of the ecliptic being given, from these observations her longitudes and latitudes, as indicated in art. 191., may be found by computation from the formulæ (A) and (B) in art. 181.

In the series of latitudes so computed there may be found two, which are equal to zero, and the moon in those points is said to be in the nodes of her orbit, for the plane of any orbit will cut the ecliptic in a line passing through the points of no latitude. The longitudes of the moon being computed for the times when the latitude is zero, it follows that the longitudes of the nodes are thereby determined: and since it is found that the nodes of the moon differ in longitude about 180 degrees, it is evident that the line joining them passes through the earth. In the series of the moon's computed latitudes there may also be two which are equal to the maximum value of the element (about 5° 9′), one on the north, and the other on the south side of the ecliptic; and this greatest latitude expresses the obliquity of the moon's orbit to that plane.

But as it is scarcely probable that, from any one of the observed right ascensions and declinations of the moon when on the meridian of the observer, the computation should give for the latitude exactly zero, methods similar to those which are put in practice for finding the place of the equinoctial point (arts. 172, 173.) may be used to determine the places of the moon's nodes. Thus let two observations be chosen, from one of which the moon's place м' is found to be south of the ecliptic, and from the other the place м is to the north: let E'E be part of the ecliptic in the heavens, M'м part of the

M

moon's apparent path; then N will be the apparent place of the ascending node: also if M'E', ME be parts of great circles passing through the moon's centre perpendicularly to the ecliptic, those arcs will be the computed latitudes, and E'E the difference of the moon's longitudes between the two

E

m

E'

N

ရာ

M/

times of observation: then E'N-NE being found by the method explained (art. 173.) for the equinoctial point, since E'N + NE is known, we have E'N and NE separately. Consequently, the longitude of the moon when at M' being known, we have the longitude of the node; and from one of the right-angled triangles, as NEM, the angle at N may be computed: this angle will be the inclination of the moon's orbit to the ecliptic.

210. Or, the inclination of the moon's orbit to the ecliptic, and the position of the line of nodes, or intersection, may be determined by the methods described in arts. 185, 186, from the longitudes and latitudes of the moon; these being deduced by computation from the formulæ (A) and (B) in art. 181., and the right ascensions and declinations being obtained from observations made at two different times.

Thus, let s (fig. to art. 184.) be the centre of the earth, and P a place of the moon at one of the times of observation: let XEY be the plane of the ecliptic, and sx', parallel to EX, the line of the equinoxes; also let x, y, z be rectangular coordinates, SA", SB", MP of P. The angle X'SM will express the longitude of the moon at P, which may be represented by L, and PSM the corresponding latitude, which may be represented by λ. Let SP be represented by r; then imagining a line as SN to be drawn through s perpendicularly to the plane of the moon's orbit, we shall have, p on this line being zero since the orbit passes through s,

x cos. NSX' + y cos. NSY + z cos. NSZ' = 0 (art. 183.) But (art. 184.)

xr cos. A cos. L.

y = r cos. λ sin. L.

z = r sin. λ;

substituting these values of x, y, z in the equation, the latter becomes

cos. A cos. L cos. NSX' + cos. λ sin. L cos. NSY

sin. A cos. NSZ=0: