For astronomical purposes and for long measures of time the Julian day is to be preferred to the year, because it avoids all irregularities of chronology. The Julian day is simply the or dinal count of mean solar days, commencing at Greenwich noon. The number for 1925, Jan. 1, mean noon is 2,424,152, follow ing the system of Joseph Scaliger, who devised it and introduced it into chronology. The number for the beginning of any other year may be derived from this, but is tabulated also in the Nautical Almanac, and elsewhere.
Rotation of the Earth.---The rotating earth is the prime time-keeper and can be readily idealized into a perfect time keeper. It is therefore desirable to examine critically how closely it approaches an absolute standard. It might be supposed that the links were few and easily dealt with. Such is not the case. They are numerous, and a fairly full outline of the treatment must suffice, referring for detail to the sources.
Let the position of the earth be defined by its principal axe of inertia (0C,OA,OB). If for the moment we suppose the earth rigid, these will be fixed in its body and evidently one may be called its polar axis (OC), while the other two (OA,OB) define the equator where they. mark determinate longitudes. The instantaneous motion of the earth may be defined in a simple manner with respect to these axes, namely if A'B'C' are their positions after an interval b t, A'B'C' may be derived from ABC by rotations about the axes OA, OB, OC. It is less simple to connect ABC, the position at the time considered, with their position A0B0C0 at an assigned moment or "epoch." It is done by means of the so-named Eulerian angles, 0, 0,4, where 0 = CC', 4 = = 18o° —CoCA. These angles serve also to express co, viz.: where A, B, C stand for the principal moments of inertia and L, M, N are the couples applied to the body about the principal axes. In the case of the earth we have very approximately, at least, symmetry about the polar axis, and the main rotation taking place about the same axis. Hence A'-"B, N-^0, and col, w2 are small. From the third equation, therefore w3= n, a constant, and the equations for cob co2 become A A )nco2 Acb,—(C—A)ncoi=M. The solution of these two consists of a free oscillation of arbitrary amplitude and phase with period 27rA (C — supplemented by any particular integral of the equa tions. Leave aside for the moment the free oscillation or "com plementary function," and direct attention to the particular integral. For simplicity consider first the sun alone. By differen tial attraction on the protuberant parts of the earth in the neighbourhood of the equator the sun produces a couple, say P, applied to the earth's body, the axis of which is in the equator and directed to a point q, perpendicular to the plane through the polar axis C and the position of the sun S. This couple is a maximum at either solstice and vanishes when the sun is at either equinox. It is easy to see that it does not change sign when the sun crosses the equator. The values of the components L, M are then L= P sin (0 —nt),M = — P cos (0 —nt), where 0 stands for the distance of the sun from the origin, measured along the equator. Now the equations for col, w2 may be written
which actually suffices for all purposes. The interpretation of this solution is that the pole C is at any moment turning in the positive sense with angular velocity P/Cn about the point s where CS meets the equator. Therefore the actual direction of the motion of C varies over two right angles, but the mean effect is a displacement of the north pole towards the vernal equinox, or what is the same thing, a regression of the equinoxes along the ecliptic. Besides its mean value, P contains also a reference to the actual positions of the sun and moon. Hence all the inequal ities that are required to specify their positions figure as periodi cal inequalities in col, W2, in special degree those of a more per manent character, such as the position of the node of the moon's orbit on the ecliptic, which will evidently affect the value of the couple contributed by the moon. All these inequalities combined constitute the nutation, an oscillation of the node of the equator upon the ecliptic, distinguishing at any date the true equinox from the mean equinox.
Since the motion of the equinox enters into every observation of position, other than merely differential ones, its determination is of the highest importance. It is an intersection of the equator with the ecliptic or plane of the earth's annual motion round the sun. This itself is in motion since the earth is perturbed by the other planets, but the motion may be calculated and allowed for. It is, however, essential first to construct the "theory of the sun," already referred to, based upon a combination of all trust worthy observations of the position of the sun with respect to the fixed stars ; and for the present object especial attention must be given to the two places in each year where the sun crosses the equator. This connects the equinoxes with the stars which can be observed with the sun, for the most part a small number of the brighter stars, and therefore not characteristic of the whole. But they again may be connected with a larger mass of fainter stars, and so on, gradually approximating to a reference to the mean of the whole body of stars which it is agreed to treat as fixed. As remarked above, it is from actual determination of the position of the equinox at different epochs that the period of the precession is deduced. It is a matter of astronomical convenience to take the sidereal day, not from a revolution with respect to the body of stars, but with respect to the true equinox. Hence the "sidereal day" as employed is not even of constant length ; to reduce it to constant length we must remove from it the nutation which has been incorporated in it. The chief amount is an i8-year fluctuation, running for half the period in one direction and half in the other and dislocating the proper zero of the day by rather more than I sec. at maximum.