On the formation of the derivatives of the lunar coördinates with respect to the elements

Abstract

In the determination of the action of the planets on the moon, it is usual to use the method of the variation of arbitrary constants. This mnethod requires the knowledge of the derivatives of the lunar coordinates with respect to the angular elements E, 7r, 0 the epochs of mean longitude, perigee and node-and also with respect to the other three elements n, e, ry -the mean motion, eccentricity and sine of the orbital inclination. If we have a theory in which the co6rdinates, so far as the actions of the sun and moon only are concerned, are expressed literally in terms of these six elements, it is a simple matter to find the derivatives. But this is practically not the case. The convergence of many of the coefficients when expressed in powers of m = n/n'the ratio of the miean motions is so slow that a literal theory with the required accuracy seems almost impossible on account of the labor required for its development; the slowness of convergence does not occur as far as powers of e, ry are concerned. The theory of Hansen is entirely numerical and therefore only the derivatives with respect to the angular elements can be obtained from it. The theory t which I hope shortly to bring to a conclusion is semi-numerical, i. e., the numerical value of mn is inserted, the other constants being left in a literal form. t In considering how this theory was to be adapted so as to be of use when considering planetary action, the only difficulty was, therefore, the formation of the derivatives with respect to n. It is this difficulty which has been solved in the following pages. The semi-numerical theory once completed, it is shown that the derivatives with respect to n can be made to depend on quadratures with respect to the time. The formulhe are put into a form ready for computation and it is shown how a simple transformation will permit us to use the advantages of canonical sets of constants without being hampered by their disadvantages for purposes of calculation.