RECENT DISCOVERIES.

83

Perturbations due to Venus.

102. After the expression for the moon's longitude had been obtained by theory, it was found that there was still a slight deviation between her calculated and observed places, and Bürg, who discovered it by a discussion of the observations of Lahire, Flamsteed, Bradley, and Maskelyne, thought it could be represented by an inequality whose period would be 184 years and coefficient 15". This was entirely conjectural, and though several attempts were made, it was not accounted for by theory.

About five years ago, Professor Hansen, of Seeberg in Gotha, having commenced a revision of the Lunar Theory, found two terms, which had hitherto been neglected, due to the action of Venus. One of them is direct and arises from a 'remarkable 'numerical relation between the anomalistic motions of the moon ' and the sidereal motions of Venus and the earth; the other is 'an indirect effect of an inequality of long period in the motions ' of Venus and the earth, which was discovered some years ago 'by the Astronomer Royal."*

6

The periods of these two inequalities is extremely long, one being 273 and the other 239 years, and their coefficients are respectively 27.4" and 23-2". These are considerable quantities ' in comparison with some of the inequalities already recognised 'in the moon's motion, and, when applied, they are found to account for the chief, indeed the only remaining, empirical 'portion of the moon's motion in longitude of any consequence; 6 so that their discovery may be considered as a practical com'pletion of the Lunar Theory, at least for the present astro'nomical age, and as establishing the entire dominion of the 'Newtonian Theory and its analytical application over that 'refractory satellite.'†

Report to the Annual General Meeting of the Royal Astronomical Society, Feb. 11, 1848.

+ Address of Sir John Herschel to the Meeting of the Royal Astronomical Society.

Motion of the Ecliptic.

103. We have seen, Art. (14), that our plane of reference is not a fixed plane, but its change of position is so slow that we have been able to neglect it, and it is only when the approximation is carried to a high order, that the necessity arises for taking account of its motion.

It has been found to have an angular velocity, about an axis in its own plane, of 48′′ in a century, and the correction thus introduced produces in the latitude of the moon a term

- cw cos(0-4),

where w is the angular velocity of the ecliptic, the angular

C

velocity with which the ascending node of the moon's orbit recedes from the instantaneous axis about which the ecliptic rotates, the longitude of this axis at time t, and 0 the longitude of the moon at the same instant.

Let TAM (fig. 11) be the position of the ecliptic at time t,
A the point about which it is turning, TA = Φ,
NM the moon's orbit, M the moon, and Mm a perpen-

dicular to the ecliptic; Tm = 0; Mm = lat. B.

=

Let APN'm' be the ecliptic after a time dt. Any point whose longitude is L may be considered as moving perpendicularly to the ecliptic with a velocity w sin(L – 4).

Hence, if i be the inclination of the orbit, and N the longitude of the node, the point N will move in the direction NP with a velocity w sin(N-4). And N' will move along PN' with a velocity w sin(N-4) coti;

Again, the point of the ecliptic 90° in advance of N, will move towards the moon's orbit with a velocity w sin(90+N−¢);

but

Now,

therefore

= ca cos(N-4) coti, Si = = co sin(N-),

d (N − p)
dt

have

SN

COS 2

=

=

cos(N-4).

sin 2

sinß = sini. sin;

cosß. SB = cosi. sin. di + sini.cos.d

85

= cw{cosi sin sin (N−4) — cos y cos(N−4)}, cos sin (0-N) and cosy = cosß cos(0-N): SB cw cos(0-4).

cosi. sin

The discovery of this term is due to Professor Hansen; its coefficient is extremely small, about 1.5"; but, being of a totally different nature from those due to successive approximations, 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 and simplicity.

na

We may obtain an approximate value of the coefficient co by substituting for it where a is the number of seconds through 2TT' which the ecliptic is deflected in one year = 0.48", and ʼn is the n number of years in which the node of the moon's orbit makes

2π

a complete revolution = 18.6; for then, is the angle described

by the node in one year; therefore,

ω:

n

na 2π

is the ratio of

d (N-), supposing to remain constant, which is nearly

dt

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

the January Examination of 1852.

Question 21, Jan. 22.

Note on the Numerical Values of the Coefficients.

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 0 and 6 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 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 o +a, a being some constant angle; then B sin0+ C sin

may

=

be written (B+C cosa) sin @ + C sina cose, or V = A + (B + C cos a) sine + C sina cose +

If now we divide the observations, as before, into two sets, corresponding to the positive and negative values of sin, the terms involving cose in each set will be as often positive as negative, and 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 sin will be cancelled, and the same process will give

Treating in the same way with respect to the angle 4, we

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

When the periods of and 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' sine, the result U will be very nearly equal to A+ Csino + &c., and from the n equations U1 = = A + C sino, +

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+Ccosa and C+ B cosa for B and C respectively.