lower than Q, and the term will have risen in importance by the integration. twice increased in value; for they increase once in forming of T de, and once again, as above, in finding t. T Since such terms occur in the development of and also dt ᏧᎾ de on account of their previously being found in u, we must examine how they appear in the differential equation that gives u, that we may recognise and retain them at the outset. Now, by referring to the last article, we see that when is very small G and G' will be of the same order; and in the order of the term will still be the same. 29. We have, therefore, the following rule: In approximating to any given order, we must, in the differential equations for u and s, retain periodical terms ONE ORDER beyond the proposed one, when the coefficient of ◊ in their argument* is nearly equal to 1 or 0; and terms in which the coefficient of the argument is nearly equal to 0, must be retained TWO ORDERS beyond the proposed approximation when they occur in T h2u3. *The angle of a periodical term is its only variable part and is called the argument. If we wished to obtain u only and not t, there would be no necessity for retaining those terms of a more advanced order in which the coefficient of 0 nearly equals 0. SECTION II. To solve the Equations to the first order. 30. We shall in this first step neglect the terms which depend on the disturbing force, i.e. those terms which contain m', for we have seen, Art. (24), that such terms will be of the second order. The differential equations may be written under the more convenient form 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. 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, sk sin (0-y) indicates motion in a plane inclined to the ecliptic at an angle tank. 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) scot MOM', = which, compared with the equation above, shews that MOM' = tan1k. 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 D approximate magnitude of the quantities e and k. By referring to Art. (22), we see that e is about and k about 1', 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. 21 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, 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 +-a) + e2 sin 2 (pt+s-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 is zero.* * 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. For, since the mean value of 0, found by rejecting the periodical terms, is pt+; if, when this vanishes, t=0, we must have = 0; therefore 0=pt+2e sin(pt — a). correct to the first order.* 19 35. We have now obtained three results, U Sv ☺v 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), * 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 abitrary 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, 0 = ε + 2e sin (ɛ — a), is the equation for determining the constant &, and in the second approximation, ε would he found from giving different values of ɛ at each successive approximation. |