MODIFICATION OF FIRST SOLUTION. 33 accuracy, and extended over several revolutions of the moon, it is found that her apse and the plane of her orbit are in constant motion. The above form of the value of u is suggested by our previous knowledge of this motion of the apse, which, as we shall see Art. (66), is connected with the value of c here introduced; and there is no doubt that Clairaut, to whom this artifice is due, was led to it by that consideration, and by his acquaintance with the results of Newton's ninth section, which, when translated into analytical language, lead at once to this form of the value of u. We might, therefore, taking for granted the results of observation, have commenced our approximations at this step, and have at once written down u = a {1+ e cos (c) —- a)}, but we should, in so doing, have merely postponed the difficulty to the next step, since there again, as we shall find, the differential equation is of the form d'u de + u = a function of 0, the correct integral of which would be, u = A cos (0 B) + and this would at the next operation bring in a term with 0 for a coefficient, which we now know must not be. We shall, there * Newton has there shewn, that if the angular velocity of the orbit be to that of the body as GF to G, the additional centripetal force is G2- F hu3, the original force being pu2. Therefore G2 fore, hereafter omit such terms as A cos() - B) altogether, and merely write u = a 38. So far, all that we know about c is that it differs from unity at most by a quantity of the first order, but its value will be more and more correctly obtained by always writing, in the successive approximations, a + ae cos (c0 − a) for the first two terms of the value of u, then the coefficient of cos(ce - a) in the differential equation must equal ae(1-c2); and this will enable us to determine c to the same order of approximation as that of the differential equation itself. See Arts. (48) and (94). 39. In carrying on the solution of S, the same difficulty arises as in u, and it will be found necessary to change it into 8 = k sin(go - y) S' 19 g being a quantity which differs from unity at most by a quantity of the first order. See Arts. (49) and (95). 40. The equation, will also be modified by this change in the value of u, Here ce and ct hold the places which and t occupied in (33); therefore or 0 = pt + 2e sin (cpt — a) to the first order, since =e to the first order. C Θ', 41. Since the disturbing forces are to be taken into account in the next approximation, we shall have to use the value of u' found in (18), which is u'a' {1 + e' cos (e')}: CONNEXION BETWEEN ' AND 0. 35 but this introduces '; we must therefore further modify it by substituting for e' its value in terms of e, and it will be found sufficient, for the purpose of the present work, to obtain the connexion between them to the first order, which may be done as follows: Let m be the ratio of the mean motions of the sun and moon, p', p their mean angular velocities; . p = mp, p't + B, pt ...... mean longitudes at time t, B being the sun's longitude when t = 0, Ꮎ true longitudes at time t, ...... 0', ૐ, But, by Art. (13), true anomaly = mean anomaly + 2e' sin (mean anomaly) + &c.; because pt = 0 -2e sin (co- a) to the first order by (40). a' {1+ e' cos (m✪ + B − )} to the first order. 42. The values of sin 2(0-0) and cos 2(0-0') can also be readily obtained to the same order: sin 2(0 – 0') = sin {(2 – 2m) 0 — 2ß — 4e' sin(m0 + B − 5)} = sin {(2 — 2m) 0 — 2ß} — 4e' sin(m0 + B − () cos {(2 − 2m) 0 – 2ẞ} = sin {(2 – 2m) 0 – 2ẞ} — 2e' sin {(2 — m) 0 — ß — Č} — 2e' cos {(2 — m) 0 − B — §} + 2e' cos {(2 – 3m) 0 – 3B + 5}. The first term of each of these is all we shall require. P SECTION III. To solve the Equations to the Second Order. 43. Let us recapitulate the results of the last approximation : u = a {1 + e cos (c) — a)}, Ꮎ 0 – 0' = (1 — m) 0 B 2e' sin(m0+B−5). -- These values must now be substituted in the expressions for 2 h2u" h2u 3 h2u3 ' h2u3 do h2u3 de do T d's (+8) de, retaining terms above the second order, when, according to the criterion of Art. (29), they promise to become of the second order after integrating. The equations (B′) and (y') of Art. (30) will then assume the forms and the integration of these will enable us to obtain u and s to the second order; after which, equation (a) of Art. (20) will give the connexion between 0 and t to the same order. 44. The quantity m'a's which we shall meet with as a coef ficient of the terms due to the disturbing force, can be replaced by m2a, m being the ratio of the mean motions of the sun and moon; for, as in Art. (24), [1 +3 cos{(2−2m) 0 — 28}] [1− 3k2 + 4k2 cos 2 (g0−y) — §m2 [1+3 cos{(2—2m) 0 — 2ß}] = a - m2e' cos(m0 + B − 5) + 3m2e cos (c0 — a) + m2e cos((2-2m — c) 0 — 2ẞ+a}. * It may perhaps be imagined, since the orbits described are not accurately ellipses, and, even if they were, a' and a would be the reciprocals of the semilatera recta, and not those of the semi-major axes, that therefore m'a'3 is only approximately equal to m3a; and that if it were required to carry our investigations to a higher order than we propose to do, it would be necessary to correct this equation. But the very fact that the moon's orbit is not an exact ellipse, shews that the yet unknown quantity a is not a definite magnitude whose value is fixed at the outset like those of m' and μ. It is a constant, it is true; but it is one whose value may be whatever we please, provided the assumption do not interfere with our continuous approximation. Now from Art. (30), where a was first introduced, we see that it is equal to h2 being itself an arbitrary constant brought in by the integration, and h2, μ this being the only relation between the quantities h and a, we are at liberty to establish any second one. The relation m'a'3 =m'a is this second assumed relation. The reasoning in the text does not prove this relation, since, as we say, it is an arbitrary one, but it suggests it. The actual value of a is now fixed, and the method of obtaining it from observation will be explained in the |