=

Lohrmann (Topographie der Sichtbaren Mondoberfläche, p. 27) gives b' (a-90°), a being equal to the distance of the moon's apparent centre from the pole (see section 11). This formula is employed in the following computations for determining points of the first order.

Since the greatest value of B' is about 1° 32', and the greatest value of B about 5o 5', it follows that b' must change sign in each lunation (see section 14).

Investigation of Libration in Longitude.

18. Libration in longitude, or the selenographical longitude of the apparent centre of the disk, is equal to the angle formed at the moon's pole between the first meridian, or that from which all selenographical longitudes are reckoned, and the circle of latitude (Moon's pole op L in fig. 6) passing through the apparent centre of the disk. This angle is equal to the Fig. 6.

selenocentric longitude of the apparent centre of the disk, reckoned on the moon's equator from the ascending node of the moon's equator on the ecliptic (which is equal to the longitude of the ascending node of the orbit +180°), minus the distance of the first meridian from the same point (see fig. 6), where 8 (N, fig. 2, p. 223) represents the ascending node of the moon's equator on the ecliptic, L the selenocentric longitude of the apparent centre σ, and L' the distance of the first meridian from 8, or its selenocentric longitude. The distance of the first meridian from the ascending node of the moon's equator on the ecliptic is, from the uniformity of the moon's rotation, at all times equal to the moon's mean longitude, minus the longitude of the ascending node of the orbit, or plus the supplement of the longitude of the ascending node. Libration in longitude vanishes when the moon is in the line of the apsides.

19. When the moon passes the point of perigee, the first meridian, 0°, of selenographical longitude appears as a straight line, which cuts the centre of the apparent disk. Libration in longitude then =0°. Should the passage of the perigee coincide with that of either node, the first meridian is projected at right angles to the equator, also a straight line; and the apparent disk is in a state of mean libration, and may be represented on the orthographical projection, subject to the necessary distortion in the regions about the limb.

The distortion on the orthographical projection arises from the greater foreshortening of objects near the limb, as seen from the earth, than the true orthographical projection will represent.

20. During the passage of the moon from perigee, at which point her motion is quickest, to apogee, where it is slowest, the motion in her orbit is slower from day to day, while her motion in rotation continues uniform; the consequence is, that while passing from perigee to mean distance the first meridian is transferred eastwardly (see fig. 7), which is inverted, where E" represents the earth, W PEo the moon's equator when she is in perigee, o being its intersection with the first meridian, W' o' E' the segment of the

Fig. 7.

W

E

moon's equator presented to the earth at a given distance from perigee, co', a radius from the moon's centre to the first meridian, the angle E" co'= the quantity gained by the axial over the orbital motion=the difference between the moon's true and mean longitudes nearly-libration in longitude, by which the western portions come into view, and the first meridian

is projected as a curve east of the centre of the apparent disk. At the point of mean distance the two motions coincide in value, but only momentarily so, the greatest libration towards the east is attained, the orbital motion becomes slower than the axial, and the first meridian returns westwardly, attaining its mean position at the passage of the apogee. In consequence of the small difference of the period of the revolution of the apsides and half that of the nodes, the equator will not appear as a straight line across the apparent disk, when the first meridian returns to its mean position, and therefore the point of 0° latitude will not be found at the centre of the apparent disk; the divergence will be greater at the end of every period either of the passage of the nodes or apsides, increasing for a period of about eighteen months, after which the divergence will decrease during another period of eighteen months, and at the end of three years (nearly) the state of mean libration will be again attained.

Libration in longitude from apogee to perigee is the opposite to that above described, from which it follows that libration in longitude changes sign every lunation.

21. The mathematical portion of this investigation may be treated under two heads, viz., the method adopted in the Nautical Almanac,' and that adopted by Lohrmann. For the method adopted in the 'Nautical Almanac ' we again refer to fig. 5, the reasoning being as follows :-

I

Since I is a very small angle, the equation tan (A'— 8)=tan (λ— 8) sec. I (see section 15) gives by a known formula of expansion A'=λ+sin 2(λ — 8) tan2 1, the rest of the terms being insignificant. The second term is A in the Nautical Almanac.' Because I is very small, and

B' is always less than I, sin

or

will be very nearly

to unity. Also

cos I COS B' because l-A', 4, and B-B' are all small arcs, we may substitute the arcs for their tangents and sines. Hence

and as the libration in longitude l'=1-1, where the moon's mean lon

=

gitude, the libration in longitude λ+A+ 7; but since, as mentioned

1

a

in section 9, -b' is to be substituted for 1, the expression becomes

22. Lohrmann, whose symbol for the moon's mean longitude is 1, and for the libration in longitude is l', gives, in 'Topographie der Sichtbaren Mondoberfläche,' p. 28, the following formula for computing the libration in longitude: l'=L-L' (see section 18 and fig. 6). Now L=270°+B−▲ and

L'=1+supp. 8 (see section 18). For the formulæ used in computing B sce section 11, and for A see section 7. These formulæ have been employed in the following computations of points of the First Order.

The principal part of the libration in longitude is 7-A (see section 9), which, besides changing sign in each lunation with respect to east and west, changes sign also with respect to north and south by the motion of the moon's apsides.

Application of the foregoing investigations to the motion on the apparent disk of the point at which the Equator intersects the First Meridian.

23. It now remains to inquire how the point of intersection of the moon's equator and first meridian will be affected by the changes in latitude and longitude which the centre of the apparent disk is perpetually undergoing ; for as only the latitude and longitude of this single point are determined by the formula for computing the librations, we do not appear to have at present the means for tracing out on the moon's disk the curves representing the moon's equator and first meridian for any other epochs than that of mean libration, when, as before mentioned, they cross the disk in two straight lines intersecting at the centre; and this inquiry is perhaps the more important as showing how necessary it is, for accurately mapping the surface, to have good determinations of points of the first order. Taking, therefore, the spot on the moon's surface at which the equator and first meridian intersect each other, we may inquire the path it will describe on the apparent disk during the changes of libration through one revolution of the nodes.

24. In fig. 9 let WENS represent a small circle concentric with the limb or margin of the apparent disk of the moon, W E being a portion of the equator,

and N S of the first meridian in mean libration at the passage of the descending node and perigee respectively, and o the point of intersection of the two(0° of latitude and longitude), and o' the position occupied by the point o by the joint effect of both librations, o E will consequently represent the greatest excursion of the point o in longitude, and o S that in latitude, the equator being projected in the curve e'o'q, and the first meridian in co'm.

The

displacement of o being in the line o o', the libration of the centre of the apparent disk will be W in longitude and N in latitude. It is easy to see that the path of the point of intersection of the equator and first meridian, a short time before and after the epoch of mean libration, will be in a very narrow ellipse, the line o' o" being the major axis, which does not, however, retain its position on the apparent disk, but revolves around the central point.

25. This ellipse opens out and undergoes changes of form proportional to the interval elapsing from the epoch of mean libration until the epoch when the greatest excursion of libration in longitude towards the east (of the point of intersection of the equator and the first meridian) coincides with the passage of the ascending node when the equator is represented as a straight line across the apparent disk and the first meridian by the curve c Em in fig. 10,

where the libration of the centre of the apparent disk is nothing in latitude, but west in longitude. When the first meridian returns to its normal position, the equator is represented by the curve E" N q (fig. 10), and the point of intersection is situated at o" (nearly); the libration of the centre in this case is nothing in longitude but south in latitude.

26. At this epoch, intermediate between two of mean libration, the path of the point of intersection of the equator and first meridian may be represented by the four diagonals, of which o' o" (fig. 10) is one, or, perhaps more correctly, by a wavy ellipse; for as the values of the two librations differ in amount, the circle W ENS is not a true representation of the excursions of the intersecting point E and W, N and S; so when the greatest deviation