242. Observation had shewn the existence of inequalities in the planetary and lunar motions which seemed to belong to two different classes. On the one hand were inequalities, such as most of those of the moon, which went through their cycle of changes in a single revolution or a few revolutions of the disturbing body; and on the other such inequalities as the secular acceleration of the moon’s mean motion or the motion of the earth’s apses, in which a continuous disturbance was observed always acting in the same direction, and shewing no signs of going through a periodic cycle of changes.
The mathematical treatment of perturbations soon shewed the desirability of adopting different methods of treatment for two classes of inequalities, which corresponded roughly, though not exactly, to those just mentioned, and to which the names of periodic and secular gradually came to be attached. The distinction plays a considerable part in Euler’s work ([§ 236]), but it was Lagrange who first recognised its full importance, particularly for planetary theory, and who made a special study of secular inequalities.
When the perturbations of one planet by another are being studied, it becomes necessary to obtain a mathematical expression for the disturbing force which the second planet exerts. This expression depends in general both on the elements of the two orbits, and on the positions of the planets at the time considered. It can, however, be divided up into two parts, one of which depends on the positions of the planets (as well as on the elements), while the other depends only on the elements of the two orbits, and is independent of the positions in their paths which the planets may happen to be occupying at the time. Since the positions of planets in their orbits change rapidly, the former part of the disturbing force changes rapidly, and produces in general, at short intervals of time, effects in opposite directions, first, for example, accelerating and then retarding the motion of the disturbed planet; and the corresponding inequalities of motion are the periodic inequalities, which for the most part go through a complete cycle of changes in the course of a few revolutions of the planets, or even more rapidly. The other part of the disturbing force remains nearly unchanged for a considerable period, and gives rise to changes in the elements which, though in general very small, remain for a long time without sensible alteration, and therefore continually accumulate, becoming considerable with the lapse of time: these are the secular inequalities.
Speaking generally, we may say that the periodical inequalities are temporary and the secular inequalities permanent in their effects, or as Sir John Herschel expresses it:—
“The secular inequalities are, in fact, nothing but what remains after the mutual destruction of a much larger amount (as it very often is) of periodical. But these are in their nature transient and temporary; they disappear in short periods, and leave no trace. The planet is temporarily withdrawn from its orbit (its slowly varying orbit), but forthwith returns to it, to deviate presently as much the other way, while the varied orbit accommodates and adjusts itself to the average of these excursions on either side of it.”[146]
“Temporary” and “short” are, however, relative terms. Some periodical inequalities, notably in the case of the moon, have periods of only a few days, and the majority which are of importance extend only over a few years; but some are known which last for centuries or even thousands of years, and can often be treated as secular when we only want to consider an interval of a few years. On the other hand, most of the known secular inequalities are not really permanent, but fluctuate like the periodical ones, though only in the course of immense periods of time to be reckoned usually by tens of thousands of years.
One distinction between the lunar and planetary theories is that in the former periodic inequalities are comparatively large and, especially for practical purposes such as computing the position of the moon a few months hence, of great importance; whereas the periodic inequalities of the planets are generally small and the secular inequalities are the most interesting.
The method of treating the elements of the elliptic orbits as variable is specially suitable for secular inequalities; but for periodic inequalities it is generally better to treat the body as being disturbed from an elliptic path, and to study these deviations.
“The simplest way of regarding these various perturbations consists in imagining a planet moving in accordance with the laws of elliptic motion, on an ellipse the elements of which vary by insensible degrees; and to conceive at the same time that the true planet oscillates round this fictitious planet in a very small orbit the nature of which depends on its periodic perturbations.”[147]
The former method, due as we have seen in great measure to Euler, was perfected and very generally used by Lagrange, and often bears his name.