XVII. Researches in physical astronomy

I SUBJOIN some further developments in the Theory of the Moon, which I have thought it advisable to give at length, in order to save the trouble of the calculator and to avoid the danger of mistake, although they may be ob­ tained with great readiness and facility by means of the Table which I have given for the purpose. While on the one hand it seems desirable to introduce into the science of Physical Astronomy a greater degree of uniformity, by bringing to per­ fection a Theory of the Moon, founded on the integration of the equations which are used in the planetary theory, it seems also no less important to complete in the latter the method hitherto applied solely to the periodic in­ equalities. Hitherto those terms in the disturbing function which give rise to the secular inequalities have been detached, and the stability of the system has been inferred by means of the integration of certain equations, which are linear when the higher powers of the eccentricities are neglected, and from consi­ derations founded on the variation of the elliptic constants. The stability of the system may, I think, also be inferred from the expres­ sions which result at once from the direct integration of the differential equa­ tions. In fact, in order that the system may be stable, it is necessary that none of the angles under the sign sine or cosine be imaginary, which terms would then be converted into exponentials, and be subject to indefinite in­ crease. In the lunar theory, the arbitrary quantities being determined with that view, according to the method here given, the angles which are intro­ duced may be reduced to the difference of the mean motions of the sun and moon, their mean anomalies and the argument of the moon’s latitude *.