proximations, it was thought desirable to examine it, and the above investigation, which was communicated to me by J. C. Adams, Esq.,* will be read with interest on account of its elegance.

With respect to the foregoing investigation, perhaps the following dN dt

remarks will not be superfluous:-The value of is not the actual

velocity of N, but its velocity relatively to the position of the node as determined when the motion of the ecliptic is neglected; its integral is therefore N the change of longitude, due to this motion, and in this integration no constant is added, zero being taken for the mean

dN value. The periodic forms of both and N shew that they oscillate dt

about mean values, the time of a complete oscillation being that required by sin(N-) and cos (N-) to go through their cycle. This relative motion of the node is analogous to that of a particle moving in a straight line under the action of a force varying directly as the distance.

Similar remarks apply to the inclination.

It must also be borne in mind that the result obtained ß is not the difference between the latitude referred to the actual ecliptic and that referred to a fixed plane; but is the difference between the calculated latitude referred to the actual ecliptic on supposition of its being fixed, and the correct latitude referred to the same actual ecliptic when its motion is taken into account.

We may obtain an approximate value of the coefficient co by substituting for it where a is the number of seconds

na
2π

through which the ecliptic is deflected in one year = 0′48′′, and n is the number of years in which the node of the moon's

2π

n

orbit makes a complete revolution = 18.6; for then, is the angle described by the node in one year; therefore,

*Now Lowndean Professor of Astronomy in the University of Cambridge.

stant, which is nearly the case; therefore,

104. When the periods of two of the terms, in the third method given in Art. (62), differ but slightly, for instance if and go through their periodic variations very nearly in the same time, the method could not then with safety be applied; for, since the same values of and would very nearly recur together during a longer time than that through which the observations would extend, the two terms would be so blended in the value of V that they would enter nearly as one term-the difference between 0 and & would be very nearly the same at the end as at the beginning of the series of observations.

105. Let us suppose the periods to be actually identical, so that = 0 + a, a being some constant angle; then

B sine+ C sin

may be written (B+C cosa) sine+C sina cose,

or

V=A+ (B+C cosa) sino + C sina cose +.......

If now we divide the observations, as before, into two sets, corresponding to the positive and negative values of

This affords the solution of a problem proposed in the Senate-House

in the January Examination of 1852. Question 21, Jan. 22.

H

sin, the terms involving cose will disappear in the summation of each set; and following the process of the method will give

B+C cosa = M suppose.

Dividing again into two sets corresponding to the positive and negative values of cose, the terms in sine will be cancelled, and the same process will give

Treating the observations in the same way with respect to the angle, we get two results,

M, N, M', N' are connected by the equation of condition,

M-M"N" - N".

When the periods of 0 and 4 are nearly, but not exactly, the same, this equation of condition will not hold, and the preceding values of B and C would not be exactly correct, but yet they would be very approximate, especially if the mean between the two values of B be taken.

106. We may also, after having taken one of these slightly erroneous values for B, make a further correction by establishing as it were a counterbalancing error in the value of C. Let B' be the value so found for B; then, from the V of each of the observations subtract the value B′ sinė,

the result U will be very nearly equal to A+C sino + &c., and from the n equations

a value C' of C will be obtained, by the rule of Art. (62), which will be very approximate, and, at the same time, agree better with B' in satisfying the equations than C itself would do.

107. When two terms whose periods are nearly equal do occur, it is plain, by examining the values of M and M', that the errors which would be committed by following the rule, without taking account of this peculiarity, would be the taking B+C cosa and C+B cosa for B and C respectively.

CHAPTER VIII.

HISTORY OF THE LUNAR PROBLEM BEFORE NEWTON.

108. The idea which most probably suggested itself to the minds of those men who first considered the motion of the moon among the stars, was that this motion is uniform and circular about the earth as a centre.

This first result is represented in our value of the longitude by neglecting all small terms and writing 0=pt.

109. It must, however, have been very soon perceived that the actual motion is far from being so simple, and that the moon moves with very different velocities at different times.

The earliest recorded attempts to take into account the irregularities of the moon's motion were made by Hipparchus, (140 B.C.) He imagined the moon to move with uniform velocity in a circle, of which the earth occupied, not the centre, but a point nearer to one side. By a similar hypothesis he had accounted for the irregularities in the sun's motion, and his success in this led him to apply it also to the moon.

It is clear that, on this supposition, the moon would seem to move faster when nearest the earth or in perigee, and slower when in apogee, than at any other points of her orbit, and thus an apparent unequal motion would be produced.