in the integration, and this in the motion of translation is

1

(e): g is the element added to y,, and this corresponds to in the motion of translation, only we must remember that g is measured wholly on the plane of maximum areas from its node, while a is measured on the fixed plane of reference as far as the node, and thence on the plane of the orbit; so that - will correspond to g: lastly, a is the longitude of the node of the plane of maximum areas, and y its inclination to the plane of reference; in the motion of translation the same quantities defining the position of the plane of the orbit are denoted by and i. In order, then, to compare the rotation formulæ with those of translation, we proceed to replace the elements a, e, e, w of the Planetary Theory by four new elements h, k, l, g, connected with the former by the relations

Let R' express the form which the function R takes when the new elements are substituted for the old: then we have

dR

In

da'

which occurs only in the formula for the epoch, the differen

tial coefficient is supposed to be taken with respect to a only in so far as a

Now if we differentiate the above expressions for h, k, l, g,

occurs explicitly in R: if we suppose a to vary also as contained implicitly in n, this differential coefficient will include the term proportional to the time, which may therefore with this understanding be omitted. In Art. 37 of the Planetary Theory this term was removed by a different transformation.

On comparing these with the rotation formulæ, it will be seen that they are identical, with the exception only that the sign of a differs from that of ; and this is accounted for by the fact that a and ♫ are measured in opposite directions. Thus, by employing elements having a like signification in the two motions of translation and rotation, we have arrived at the very remarkable result that the complete solution of the problem of Planetary perturbation, whether in the motion of translation or of rotation, is expressed by the above simple formula.

PART II.

APPLICATION OF PRECEDING RESULTS.

PRECESSION AND NUTATION.

23. THE formulæ obtained in the first part are sufficient completely to determine the motion of a planet or other rigid body about its centre of gravity. They are perfectly rigorous, subject only to the hypothesis that the disturbing bodies may be supposed to attract as if condensed into their respective centres of gravity; a hypothesis admissible if these bodies are either very distant or nearly spherical in form. But though the formulæ are exact, they can be integrated only by approximation. We propose, therefore, in the second part to restrict ourselves to the particular case of the Earth, taking advantage of such of the results of observation as may be required to enable us to approximate. In order to treat the problem fully we shall consider, first, the motion of the axis of rotation in the Earth itself, with the velocity of rotation about it; secondly, the motion of this axis in space. The first is of special interest, since any change in the position of the axis in the Earth, were such change possible, would affect the permanence of terrestrial latitudes; any change in the velocity of rotation would affect the length of the day. The second is of great importance to astronomers, since it establishes the fact that the first point of Aries, or vernal equinox, to which they are accustomed to refer celestial longitudes, is not a fixed point.

Stability of the axis of rotation in the Earth and
of the velocity about it.

24. We proceed then, first, to consider the motion of the axis of rotation within the Earth, and shall be able to shew that, in so far as it depends upon the attractions of other bodies, this axis can never separate appreciably from the axis of figure, and that the velocity about it must always remain appreciably constant.

2

2

A3w ̧2 + B2∞ ̧2 + C2w ̧2 = k2.

Eliminating w, from these equations,

A (C−A) w ̧2+B (C − B) w22 = Ch — k2.

Now if C be either the greatest or least principal moment, both the terms of the left-hand member of this equation must always retain the same sign; and if we take Cabout the Earth's axis of figure it will be the greatest and this sign will be positive. We may therefore write

Ch― k2 = (C−A) (C – B) e3 :

2

thus A (C-A) w ̧2 + B ( C − B) w ̧2 = ( C — A) ( C— B) e2.

If we neglect the disturbing force, h, k and therefore e are constant, and the value of e will be found by substituting