mean longitude of the ascending node and the inclination of the orbit at the beginning of each year, and the secular inequalities of these two quantities are given. Thus the mean motions are given, and the true motions are found by applying the inequalities, the numerical values of which are called equations: for, in astronomy, an equation signifies the quantities that must be added or taken from the mean results, to make them equal to the true results.

The mean motion and equation of the centre are computed from Kepler's problem; the motions of the nodes and perihelia, the secular inequalities of the elements, and the periodic inequalities, are computed from the formulæ determined by the problem of three bodies.

Method of correcting Errors in the Tables.

662. As astronomical tables are computed from analytical formulæ, determined on the principles of universal gravitation, no error can arise from that source; but the elements of the orbit, though determined with great accuracy by numerous observations, will lead to errors, because each element is found separately; whereas these quantities are so connected with each other, that a perfectly correct value of one, cannot be determined independently of the others. For example, the expressions in page 261 show, that the eccentricity depends on the longitudes of the perihelia, and the longitude of the perihelion is given in terms of the eccentricities. A reciprocal connexion exists also between the inclination of the orbit and the longitude of the nodes. Hence, in an accurate determination of the elements, it is necessary to attend to this reciprocal connexion.

The tables are computed with the observed values of the elements; an error in one of the elements will affect every part of the tables, and will be perceived in the comparison of the place of the body derived from them, with its place determined by observation. Were the observation exact, the difference would be the true error of the tables; but as no observation is perfectly accurate, the comparison is made with 1000, or even many thousands of observations, so that their errors are compensated by their numbers.

The simultaneous correction is accomplished, by comparing a longitude of the body derived from observation, with the longitude corresponding to the same instant in the tables.

Suppose the tables of the sun to require correction, and let E represent the error of the tables, or the difference between the longitude of the tables and that deduced from observation, at that point of the orbit where his mean anomaly is 198°. There are three sources from whence this error may arise, namely, the mean longitude of the perigee, the greatest equation of the centre, and the epoch of the tables; for, if an error has been made in computing the initial longitude, it will affect every subsequent longitude. Now, as we do not know to which of these quantities to attribute the discrepancy, part of it is assumed to arise from each. Let P be the unknown error in the longitude of the perigee, e that in the greatest equation of the centre, and e that in the epoch. In order to determine these three errors, let us ascertain what effect would be produced on the place of the sun, where his mean anomaly is 198°, by an error of 60′′ in the longitude of the perigee. As the mean anomaly is estimated from perigee, a minute of change in the perigee will produce the change of one minute in the mean anomaly corresponding to each longitude; but the table of the equation of the centre shows that the change of 60" in the mean anomaly at that part of the orbit which corresponds to 198° produces an increment of 1.88 in the equation of the centre; and as that quantity is subtractive at that part of the orbit, the true longitude of the sun is diminished by 1".88; hence if 60" produce a change of 1".88 in the true longitude, the error P will produce a change of

Again, if we suppose the greatest equation of the centre to be augmented by any arbitrary quantity as 17.18, it is easy to see by the tables that the equation of the centre at that point of the orbit where the mean anomaly is 198° is increased by 5".1; whence the true longitude is diminished by 5".1. Thus, if 17".18 produce a change of 5".1 in the true longitude, the error e will produce the change

Hence the sum of the three errors is equal to E, the error of the tables

+0".3133 P. 0".2969 e E.

This is called an equation of condition between the errors, because it expresses the condition that the sum of the errors must fulfil.

As there are three unknown quantities, three equations would be sufficient for their determination, if the observations were accurate; but as that is not the case, a great number of equations of condition must be formed from an equal number of observed longitudes, and they must be so combined by addition or subtraction, as to form others that are as favourable as possible for the determination of each element. For example, in finding the value of P before the other two, the numerous equations must be so combined, as to render the coefficient of P as great as possible; and the coefficients of e and e as small as may be; this may always be accomplished by changing the signs of all the equations, so as to have the terms containing P positive, and then adding them; for some of the other terms will be positive, and some negative, as they may chance to be; therefore the sum of their coefficients will be less than that of P.

Having determined this equation, in which P has the greatest coefficient possible, two others must be formed on the same principle, in which the coefficients of the other two errors must be respectively as great as possible, and from these three equations values of the three errors will be easily obtained, and their accuracy will be in proportion to the number of observations employed. These values are referred to the mean interval between the first and last observations, supposing them not to be separated by any great length of time, and that the mean motion is perfectly known. Were it not, as might happen in the case of the new planets, an additional error may be assumed to arise from this source, which may be determined in the same manner as the others. in astronomical tables was employed by Mayer, in computing tables of the moon, and is applicable to a variety of subjects.

This method of correcting errors

663. The numerous equations of condition of the form E+0.3133 P + 0.2969 e,

may be combined in a different manner, used by Legendre, called the principle of the least squares.

If the position of a point in space, is to be determined, and if a series of observations had given it the positions n, n', n", &c., not differing much from each other, a mean place M must be found, which differs as little as possible from the observed positions n, n', n",

&c.: hence it must be so chosen that the sum of the squares of its distances from the points n, n', n', &c., may be a minimum; that is,

(Mn)2 + (Mn')2 + (Mn")2 + &c. = minimum.

A demonstration of this is given in Biot's Astronomy, vol. ii.; but the rule for forming the equation of the minimum, with regard to one of the unknown errors, as P, is to multiply every term of all the equations of conditions by 0".3133, the coefficient of P, taken with its sign, and to add the products into one sum, which will be the equation required. If a similar equation be formed for each of the other errors, there will be as many equations of the first degree as errors; whence their numerical values may be found by elimination.

It is demonstrated by the Theory of Probabilities, that the greatest possible chance of correctness is to be obtained from the method of least squares; on that account it is to be preferred to the method of combination employed by Mayer, though it has the disadvantage of requiring more laborious computations.

The principle of least squares is a corollary that follows from a proposition of the Loci Plani, that the sum of the squares of the distances of any number of points from their centre of gravity lis a minimum.

664. Three centuries have not elapsed since Copernicus introduced the motions of the planets round the sun, into astronomical tables about a century later Kepler introduced the laws of elliptical motion, deduced from the observations of Tycho Brahe, which led Newton to the theory of universal gravitation. Since these brilliant discoveries, analytical science has enabled us to calculate the numerous inequalities of the planets, arising from their mutual attraction, and to construct tables with a degree of precision till then unknown. Errors existed formerly, amounting to many minutes; which are now reduced to a few seconds, a quantity so small, that a considerable part of it may perhaps be ascribed to inaccuracy in observation.

BOOK III.

CHAPTER I.

LUNAR THEORY.

665. THERE is no object within the scope of astronomical observation which affords greater variety of interesting investigation to the inhabitant of the earth, than the various motions of the moon : from these we ascertain the form of the earth, the vicissitudes of the tides, the distance of the sun, and consequently the magnitude of the solar system. These motions which are so obvious, served as

a measure of time to all nations, until the advancement of science taught them the advantages of solar time; to these motions the navigator owes that precision of knowledge which guides him with well-grounded confidence through the deep.

Phases of the Moon.

666. The phases of the moon depend upon her synodic motion, that is to say, on the excess of her motion above that of the sun. The moon moves round the earth from west to east; in conjunction she is between the sun and the earth; but as her motion is more rapid than that of the sun, she soon separates from him, and is first seen in the evening like a faint crescent, which increases with her distance till in quadrature, or 90° from him, when half of her disc is enlightened as her elongation increases, her enlightened disc augments till she is in opposition, when it is full moon, the earth being between her and the sun. In describing the other half of her orbit, she decreases by the same degrees, till she comes into conjunction with the sun again. Though the moon receives no light from the sun when in conjunction, she is visible for a few days before and after it, on account of the light reflected from the earth.