The latitude s of the moon can never exceed the inclination of the orbit to the ecliptic; but this inclination is of the first order, therefore s is at least of the first order and s2 may be neglected. whence u = {1 + e cos(-a)}, or, writing a for and ре μ (U1), (S1), e, a, k, y being the four constants introduced by integration. 31. These results are in perfect agreement with what rough observations had already taught us concerning the moon's motion Art. (22); for u = a {1 + e cos(0 − a)} represents motion in an ellipse about the earth as focus. Again, s = k sin (0− y) indicates motion in a plane inclined to the ecliptic at an angle tank. For, if TOM' be the ecliptic, (fig. 7) then M the moon's place, MM' an arc perpendicular to the ecliptic, SOLUTION TO FIRST ORDER. 29 and if To be taken equal to y, and OM joined by an arc of great circle, we have or sin OM' = tan MM' cot MOM'; sin (0 − y) = s cot MOM', which, compared with the equation above, shews that Therefore, the moon is in a plane passing through a fixed point O and making a constant angle with the ecliptic; or, the moon moves in a plane. 32. What the equations can not teach us, however, and for which we must have recourse to our observations, is the approximate magnitude of the quantities e and k. By referring to Art. (22), we see that e is about and k about, that is, both these quantities are of the first order. Their exact values cannot yet be obtained: the means of doing so from multiplied observations will be indicated further on. The values of a and y introduced in the above solutions are respectively the longitude of the apse and of the node. 33. Lastly, to find the connexion between t and 0, the equation (a) becomes, making T = 0, dt 1 = = do hu2 ha2 {1 + e cos(0 − a)}2 Now this is the very same equation that we had connecting ́t and in the problem of two bodies, Art. (12), as we ought to expect, since we have neglected the sun's action. Therefore, if p be the moon's mean angular velocity, we should, following the same process as in the article referred to, arrive at the result 0 = pt + + 2e sin (pt + e − a) + že2 sin2 (pt + e − a) + which is correct only to the first order, since we have rejected some terms of the second order by neglecting the disturbing force. 34. The arbitrary constant e, introduced in the process of integration, can be got rid of by a proper assumption: this assumption is, that the time t is reckoned from the instant when the mean value of 0 is zero.* For, since the mean value of 0, found by rejecting the periodical terms, is pt+; if, when this vanishes, t = 0, we must have ε = 0; therefore 0 = pt + 2e sin (pt — a) . correct to the first order.† 19 19 35. We have now obtained three results, U1, S1, ☺, as solutions to the first order of our differential equations, and we must employ them to obtain the next approximate solutions: but before U, and S can be so employed they must be slightly modified, in such a manner however as not to interfere with their degree of approximation. The necessity for such a modification will appear from the following considerations: Suppose we proceed with the values already obtained; we have, by Art. (23), *When a function of a variable contains periodical terms which go through all their changes positive and negative as the variable increases continuously, the mean value of the function is the part which is independent of the periodical terms. † We shall also employ this method of correcting the integral in our next approximation to the value of 0 in terms of t; and if we purposed to carry our approximations to a higher order than the second, we should still adopt the same value, that is, zero, for the arbitrary constant introduced by the integration. To shew the advantage of thus correcting with respect to mean values: suppose we reckoned the time from some definite value of 0, for instance when 0=0; then, in the first approximation, MODIFICATION OF FIRST SOLUTION. 31 and this being substituted in the differential equation (B') of Art. (30), gives Our first approximate value u = a {1 + e cos(0 − a)} is thus corrected by a term which, on account of the factor 0, admits of indefinite increase, and thus becomes ultimately a more important term than that with which we started as being very nearly the true value, and which is confirmed as such by observation (22): for the moon's distance, as determined by her parallax, is never much less than 60 times the earth's radius; whereas this new value of u, when is very great, would make the distance indefinitely small; and, on the same principle, we see that any solution, which comprises a term of the form A sin(-a), cannot be an approximate solution except for a small range of values of 0. Such terms if they really had an existence in our system, must end in its destruction, or at least in the total subversion of its present state; 'but when they do occur, they have their origin, not in the nature of the differential equations, but in the imperfection of our analysis, and in the 'inadequate representation of the perturbations, and are to be got rid of, or rather included in more general expressions of a periodical nature, by a more refined investigation than that which led us to them. The nature 'of this difficulty will be easily understood from the following reasoning. 'Suppose that a term, such as a sin (40 + B), should exist in the value of u, in which A being extremely minute, the period of the inequality ' denoted by it would be of great length; then, whatever might be the value 'of the coefficient a, the inequality would still be always confined within 6 6 6 ́ certain limits, and after many ages would return to its former state. 'Suppose now that our peculiar mode of arriving at the value of u, led 'us to this term, not in its real analytical form a sin (40 + B), but by the is the equation for determining the constant ɛ, and in the second approximation, & would be found from 0+2e sin(ε - a) + 1⁄2 e2 sin 2 (ε − α) + . ε giving different values of ɛ at each successive approximation. 6 'way of its development in powers of 0, a ± ß0 + y2+ &c.; and that, not ' at once, but piecemeal, as it were; a first approximation giving us only 'the term a, a second adding the term ẞ0, and so on. If we stopped here, 'it is obvious that we should mistake the nature of this inequality, and that a really periodical function, from the effect of an imperfect approximation, 'would appear under the form of one not periodical....................... .These terms ' in the value of u, when they occur, are not superfluous; they are essential 'to its expression, but they lead us to erroneous conclusions as to the 'stability of our system and the general laws of its perturbations, unless we keep in view that they are only parts of series; the principal parts, 'it is true, when we confine ourselves to intervals of moderate length, 'but which cease to be so after the lapse of very long times, the rest of 'the series acquiring ultimately the preponderance, and compensating the 'want of periodicity of its first terms.'-SIR JOHN HERSCHEL, Encyclopædia Metropolitana-PHYSICAL ASTRONOMY, p. 679. 36. To extricate ourselves from this difficulty, and to alter the solution so that none but periodical terms may be intro duced, let us again observe that the equation 1 μ + u = = α [h] of Art. (30), which gave the solution U, and thus led to the difficulty, is only an approximate form of the first order of the exact equation (B') of the same article. Any value of u, therefore, d'u de which satisfies the approximate equation + u = a to the first order, and which evades the difficulty mentioned above, may be taken as a solution to the same order of the exact equation (B'). Such a value will be u = a {1+ e cos (c0 — a)}....... provided 1 – c2 be of the first order at least, for then ... U1, 37. The observations hitherto made to check our approximations were extremely rough (22), and carried on only for a short interval; but when they are made with a little more |