as to include the terms that involve e2. An extension requiring no new principle or process, but merely a greater length of calculation. By taking account of the terms that involve e2, e3, &c. we obtain, after integrating the first equation (see p. 144.), a more correct value of u. The same end is attained, when the eccentricity of the Solar Orbit and the inclination of the planes of the two orbits are introduced, as conditions of the problem. But it is not merely a more correct value of u that is obtained by the introduction of these two latter conditions. A rise is given to new equations, (see Astronomy, pp. 150, &c.). We can never, by calculation, account for the annual equation (see Astronomy, p. 328.) whilst u ́ is considered equal to ; for that equation depends on 1 a the eccentricity é: nor can we, as it is plain, compute either the regression of the nodes, or the variation of the inclination of the plane of the orbit, whilst, by assuming a certain value for u, we tacitly assume the body to have no latitude. These points, which are now only slightly glanced at, will be more fully discussed in a subsequent part of this Work. CHAP. X. On the Form of the Differential Equation, when the Approximation includes Terms that involve e2. The Error, in the Computed Quantity of the Apogee, the same as before, and very little lessened by taking account of Terms involving e3. THE analytical values of P, T, remain the same as in the former case: the first alteration necessary to be made is in the values of sin. 2 w, cos. 2 w Values of cos. 2, sin. 2 w, (see pp. 129. 139.). Since the square of e is to be retained, we shall have (see p. 139.), C, which ought to appear in the denominators of the second and third term, being supposed = 1. We have therefore, instead of the former equation (see p. 140.), for deducing sin. 2 and cos. 2 w, this (the solar orbit being still considered circular), - v = v v (1-m) + 2e m sin. со and thence we have 3 e2 m 4 sin. 2 cv, cos. 2 = cos. [2 v. (1 m) + 4 em sin. c v] 3 em + sin. 2 cv. sin. [2 v (1 m) + 4 em sin. cv], 4 the cosine of sin. 2 cv being nearly 1, and the sine of = 3 e2 m the same quantity being nearly the p. 104.) quantity itself (see Trig. Now the cosine and sine of 2v (1 m) + 4 em sin. cv, are the cosine and sine of 2 w in the former case* (see p. 140.): consequently, since sin. 2 cv. sin. (2v — 2 m v) : = [cos. (2 v− 2 m v − 2 cv) — cos. (2 v − 2 mv + 2 cv)] now, instead of cos. 4 em . sin. cv, we have written 1 (rad.) from the smallness of 4 em. sin. cv: but a nearer value is consequently, the succeeding values of cos. 2 w, which are used in the text, depend on the rejection of terms involving (em), which, in the Lunar Theory, is very small. and similarly, sin. 2 w = sin. (2 v − 2 m v) – 2 m e . sin. (2 v — - 2 m v But, since it is not intended to include terms that involve e3, &c., we have, as before, [cos. (2v-2mv-cv) — cos. (2v −2mv+cv)} 4 e2. cos. cv sin. cv. sin. 2 w = 2 e2 [sin. 2 c v. sin. (2 v 2 m v)] * This method is easily extended to find cos. 2 w, sin. 2 w, when the terms that involve e3, &c. are taken account of: for if 2 w = 2w+ 2e3 sin. 2 x, then cos. 2 w' cos. 2 w - e3 [cos. (2 w — 2 x) — cos. (2 ∞ + 2 x)]. Again, if 2" 2 w' - 2. sin. 2 z, cos. 2 w" cos. 2 w' e4. [cos. (2 w' - 2 z) — cos. (2 w' + 2 z)]. = e2 [cos. (2 v — 2 m v — 2 c v) — cos. (2 v — 2 m v + 2 c v)]. Hence, 2mv - cv) 2 mv + cv) Here we see that the extending the approximation, so as to include those terms that involve e2, adds to the value of the preceding quantity three new terms, and consequently must add three new equations (see Astronomy, pp. 324, &c.) for determining and correcting the value of u. The arguments of the new equations (one of which has occurred before) are 2v-2m v 2 cv, 2v2 mv + 2 cv, and 2 v P Value of (see pp. 129, 138, 143.) 1 h2u2 m' u's = (1-3 cos. 2) P hu2 2. (1 + e2 + e. cos. c v) -3 2 m v; * In deducing the progression of the Lunar Apogee, it is necessary to compute to the greatest exactness the coefficient of cos. cv: now in the expansion of (1+e cos. cv), the term involving the cube of cos. cv |