obtained in terms of the fourth t; that is, the radius vector, longitude, and latitude would be known corresponding to a given time. 21. But the integration has never yet been effected, except for particular values of P, T, and S; and the method which we are in consequence forced to adopt, is that of successive approximation, by which the values of u, 0, and s are obtained in a series, the terms proceeding according to ascending powers of small fractions, some one being chosen as a standard with which all others are compared, and the order of the approximation is esteemed by the highest power of the small fractions retained. It is usual to consider consequently as a small fraction of the first order, of 2% = 7t0 is second...... third and so on, other fractions being considered as of the 1st, 2nd, &c. orders, according as they more nearly coincide with 20, 4to, &c. 22. It is necessary therefore, before we can approximate at all, that we should have a previous knowledge (a rough one is sufficient) of the values of some of the quantities involved in our investigations; and for this knowledge we must have recourse to observation. We shall therefore assume as data the following results of observation: (1) The moon moves in longitude about thirteen times as fast as the sun. The ratio of the mean motions in longitude represented by m is therefore about, and may be considered as of the 1st order.* *This approximate value of m is easily obtained;-the moon is found to perform the tour of the heavens, returning to the same position among the fixed stars, in about 273 days; the sun takes 365 days to accomplish the same journey. (2) The sun's distance from the earth is about 400 times as great as the moon's distance. = Hence the ratio of the mean distances is of the second order.* (3) The eccentricity e' of the elliptic orbit which the sun approximately describes about the earth is about, and this, approaching nearer in value to than to bō, will be considered as of the 1st order. (4) During one revolution, the moon moves pretty accurately in a plane inclined to the plane of the ecliptic at an angle whose tangent is about, and therefore of the 1st order.† (5) Its orbit in this plane is very nearly an ellipse having the centre of the earth in its focus, and whose eccentricity is about equal to our standard of small fractions of the 1st order, viz.; and this will also be very nearly true of the projection of the orbit on the plane of the ecliptic.‡ * The distances of the luminaries may be calculated from their horizontal parallaxes, found by observations made at remote geographical stations. + That the moon's orbit during one revolution is very nearly a plane inclined as we have stated, will be found by noting her position day after day among the fixed stars; and the sun's path having previously been ascertained in a similar way, the rules of Spherical Trigonometry will easily enable us to verify both statements. The elliptic nature and value of the eccentricity of the moon's orbit may be found by daily observation of her parallax, whence her distance from the earth's centre may be determined: corresponding observations of her place in the heavens being taken, and corrected for parallax to reduce them to the earth's centre, will determine her angular motion. Lines proportional to the distances being then drawn from a point in the proper directions, the extremities mark out the form of the moon's orbit. A similar method applied to observations of the diameter of the sun will determine the eccentricity of its orbit. Therefore the orbit of G about S is very approximately an ellipse with S in the focus, and the plane of this ellipse is, as far as our investigations are concerned, a fixed plane when S is fixed. This fixed plane is called the plane of the ecliptic, or simply the ecliptic. 16. A plane through the earth's centre parallel to the ecliptic will be the plane of reference we require (14) and will become a fixed plane when we bring the earth's centre to rest, the ecliptic then making small monthly oscillations from one side to the other of our fixed plane. 17. The sun will have a latitude always of the same name as that of the moon, and deducible from it, when M M ES, EM, and the ratio of the masses of the earth and moon are known. For if S'EM' be this fixed plane through E, and S', G', M', the projections of S, G, M, To calculate the values of P, T, S. 23. We are now in possession of the data requisite for beginning our approximations, and we shall proceed to the determination of the values of P, T, and S in terms of the coordinates of the positions of the sun and moon. Let S, E, M be the centres of the sun, earth, and moon, E m', E, M their masses, E', M', the projections on the plane of the ecliptic, MGM'tan's moon's latitude, E' the direction of the first point of Aries, SG=r' =1; <YE'S=0' = longitude of sun, == u' M'E' =r; TE'M' == longitude of moon, 1 И 0 .. SE'M' = 0 — 0' = difference of longitude of sun and moon. The forces we have to take into account are, according to Art. (9), the forces which act directly on M, and forces equal and opposite to those which act on E;-these last being applied to the whole system so that E may be a fixed point. Attraction of S m' upon M= SM2 in MS, equivalent to |