communication, and scarcely any important advance was made by the one which was not at once utilised and developed by the other.
The central problem was that of the secular alterations in the elements of a planet's orbit regarded as a varying ellipse. Three of these elements, the axis of the ellipse, its eccentricity, and the inclination of its plane to a fixed plane (usually the ecliptic), are of much greater importance than the other three. The first two are the elements on which the size and shape of the orbit depend, and the first also determines (by Kepler's Third Law) the period of revolution and average rate of motion of the planet;[1] the third has an important influence on the mutual relations of the two planets. The other three elements are chiefly of importance for periodical inequalities.
It should be noted moreover that the eccentricities and inclinations were in all cases (except those specially mentioned) considered as small quantities; and thus all the investigations were approximate, these quantities and the disturbing forces themselves being treated as small.
245. The basis of the whole series of investigations was a long paper published by Lagrange in 1766, in which he explained the method of variation of elements, and gave formulae connecting their rates of change with the disturbing forces.
In his paper of 1773 Laplace found that what was true of Jupiter and Saturn had a more general application, and proved that in the case of any planet, disturbed by any other, the axis was not only undergoing no secular change at the present time, but could not have altered appreciably since "the time when astronomy began to be cultivated."
In the next year Lagrange obtained an expression for the secular change in the inclination, valid for all time. When this was applied to the case of Jupiter and Saturn, which on account of their superiority in size and great distance from the other planets could be reasonably treated as forming with the sun a separate system, it appeared that the changes in the inclinations would always be of a periodic nature, so
- ↑ This statement requires some qualification when perturbations are taken into account. But the point is not very important, and is too technical to be discussed.
