42. The values of sin2 (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+ß− Č) cos {(2−2m) 0–2ß} = = · sin {(2 — 2m) 0 — 2ß} – 2e′ sin {(2 — m) 0 — ß — 5} Similarly, cos 2 (0 — 0′) = cos {(2 — 2m) 0 — 2ß} 1 – 2e' cos {(2 — m) 0 − B − (} + 2e' cos {(2 − 3m) 0 − 3ẞ + C}. The first term of each of these is all we shall require. SECTION III. To solve the Equations to the Second Order. 43. Let us recapitulate the results of the last approximation. 0 — 0' = (1 − m) 0 − B − 2e' sin (m0 + B − 5). These values must now be substituted in the expressions for P T Ps-S h*u*, h2u3, T du T ds /d2u hu hu3 do' hu3 do' do 3 (+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 (7') 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. m'a's 44. The quantity ha which we shall meet with as a coefficient 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. So long as we neglected the disturbing force, h and a had determinate values:-they belonged to the ellipse which formed our first imperfect solution, and would therefore be known from the circumstances of motion in that ellipse at any instant, h being double the area described in a unit of time, and a the reciprocal of the semi-latus rectum. It would consequently be impossible to assume any arbitrary connexion between them. But, when we proceed to a second approximation and introduce the disturbing force, there is no longer a determinate ellipse to which the h and a apply: the equation μ=h'a of Art. (30) merely shews that a and h must refer to some one of the instantaneous ellipses which the moon could describe about the earth if the disturbance were to cease, and we are at liberty to select any one of these which will allow us to proceed with our approximation. The particular ellipse will be determined by the assumed m'a's relation = m'a, and the selection is suggested and h'a but, since the instantaneous ellipses are nearly circles, we = a(— 1m2 {1+3e' cos(m0+ B − 5)} {1 − 3e cos (c0 − a)} [1 +3 cos {(2-2m) 0-26}] [1–3 k2+3k2 cos2 (g0−y) — {m2[1+3 cos{(2−2m) 0–28}] = a - 3m2e' cos (m0 + B − 5) + 3m2e cos (c) — a) +2me cos ((2-2m −c) 0-2ẞ+a}. The last three terms are retained, though of the third order, according to Art. (29). The first of the three will not rise in importance in the value of u, but it is retained for its subsequent use in finding t, when it will become of the second order. The other terms of the third order, which arise in the development of the expression, are neglected, as the coefficients of in their arguments are neither small 3m'a (1+e' cos (m0+B−()}3 3m2 (1+3e' cos(m0+ B− })} sin {(2 — 2m) 0 — 2B) [1-4e cos(c0—a)+10e2 cos2(c0—a)] sin{(2—2m) 0—2ß} - §m2 {1+3e' cos (m0+ B− })} [sin((2-2m) 0-2ẞ} - 2e sin {(2 - 2m −c) 0 — 2B+a} + že2 sin {(2 – 2m — 2c) 0 − 2B+2a}] (sin{(2—2m)0—2ß} — 2e sin {(2—2m−c)0−2B+a} (+ §e2 sin {(2 − 2m — 2c) 0 −2B + 2a}. We have, in the course of the reduction, dropped those terms which, according to Art. (29), could not produce important terms in the resulting value either of u or of t. The last term, though of the fourth order, is retained because 2- 2m - 2c is small. T du h'u de= (previous expression) {— aec sin (cơ — a)} h2u do Here the denominator is of the first order, and cannot be further simplified without a more accurate knowledge of the value of c. We shall find in the next value of u, Art (48), that 1-c is of the second order, and as this result is obtained independently of the term we are here considering, which is only retained for the sake of finding t, there is no impropriety in anticipating thus far in order to simplify this coefficient, which then becomes T น [] d0 = 4m2 cos {(2 — 2m) 0 – 2,8} — 3m3e cos {(2 — 2m — c) 0 − 2B + a} - 15 me3 cos {(2 – 2m — 2c) 0 − 28 +2a}. Also, by Art. (36), d'u do +u= a+ quantities of the second order, +8= small quantity of the second order at least; + d2s do - 15 me a cos ((2-2m-2c) 0-2B+2a}, + 8) de=0, to the third order. и |