differential equations. The integrations being performed as before will give the values of u, s, and t in terms of ◊ to the third order, and from these, as in Arts. (51), (52), and (53), may be obtained u, s, and 0 in terms of t. 57. More approximate values of c and g are obtained at the same time, by means of the coefficients of cos (c✪ − a) and sin (gy) in the differential equations, (see Appendix, Arts. 94 and 95). 58. The values to the fourth order are then obtained from those of the third by continuing the same process, and so on to the fifth and higher orders; but the calculations are so complex that the approximations have not been carried beyond the fifth order, and already the value of 0 in terms of t contains 128 periodical terms, without including those due to the disturbances produced by the planets. The coefficients of these periodical terms are functions of m, e, e', a c, g, k, and are themselves very complicated under their literal forms that of the term whose argument is twice the difference of the longitude of the sun and moon, for instance, is itself composed of 46 terms, combinations of the preceding constants. See Pontécoulant, Système du Monde, tom. IV. p. 572. CHAPTER V. NUMERICAL VALUES OF THE COEFFICIENTS. 59. Having thus, from theory, obtained the form of the developments of the coordinates of the moon's position at any time, the next necessary step is the determination of the numerical values of the coefficients of the several terms. We here give three different methods which may be employed for that purpose, and these may, moreover, be combined according to circumstances. 60. First method. By particular observations of the sun and moon (i.e. by observations made when they occupy particular and selected positions), and also by observations separated by very long intervals, such, for instance, as ancient and modern eclipses, the values of the constants p, m, a, B, y, 5, which enter into the arguments, and of the additional ones which enter into the coefficients of the terms in the previous developments, may be obtained with great accuracy, and by their means, the coefficients themselves; c and g being also known in terms of the other constants. These may properly be called the theoretical values of the coefficients, the only recourse to observation being for the determination of the numerical values of the elements. 61. Second method. Let the constants which enter into the arguments be determined as in the first method; and let a large number of observations be made, from each of which a value of the true longitude, latitude, or parallax is ob tained, together with the corresponding value of t reckoned from the fixed epoch when the mean longitude is zero. Let these corresponding values be substituted in the equations, each observation thus giving rise to a relation between the unknown constant coefficients. A very great number of equations being thus obtained, they are then, by the method of least squares or some analogous process, reduced to as many as there are coefficients to be determined. The solution of these simple equations will give the required values. This method, however, would scarcely be practicable in a high order of approximation. For instance, in the fifth order, as stated in Art. (58), each of the numerous equations would consist of 130 terms, and these would have to be reduced to 129 equations of 130 terms each. 62. Third method. When the constants which enter into the arguments have been determined by the first method, we may obtain any one of the coefficients independently of all the others by the following process, provided the number of observations be very great. Let the form of the function be V=A+B sin0+ C sino + &c., and let it be required to determine the constants A, B, C, &c. separately; 0, 4, &c. being functions of the time. Let the results of a great number of observations corresponding to values 01, 02, 03, &c., 41, 42, 43, &c., be V1, V2, V1 &c.; so that V1 = A + B sin 0, + C sinø, +&c., 1 29 Now, n being very great, we may assume that the sum of therefore 1 A = n n which determines the non-periodic part of the function. To determine B. Let the observations be divided into two sets separating the positive and negative values of sin ; then the other periodical terms, not having the same period, may be considered as cancelling themselves in adding up the terms of each set. Let there be r terms in the first set and s terms in the second, and let V', V", ...... V' be the values of V corresponding to positive values of sin 0, which values we may assume to be uniformly distributed from sin0 to sin, and therefore to be sin de, sin 280, sinr.80, where r.80 =T. 119 1119 ...... ...... And, again, let V, V, V V1, be the values of V corresponding to the negative values of sine, viz.; sin (— ▲¤), sin(-240), ...... sin(-s.A), where s.AT. V' = A + B sin de +Csino' +...,|V, A-B sin A0 Then, +C sino, +.... V" = A + B sin 2.80+ C sino" +..., VA - B sin 2.40+C sin &,, +..... therefore .... V' = A + B sinr.80 + C sino +...; V1 = A – B sins.▲0 +C sin &, + therefore V'+V" + ...+V'=r.A+ B2" (sin 0) V, +V+...+V=8. A-BY," (sin0) and in a similar manner may each of the coefficients be independently determined.* For further remarks on this method, see Appendix, Art. (104). sinode for π * If r and are not sufficiently great to allow us to substitute " sine.de, we must proceed as follows: V'+V"+ • + V* = rA + B (sin d0+ sin 280 + + sin rò0) ...... |