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302 in subsequent researches, it may be worth while to attempt to give a sketch of it.

If perturbations are ignored, a planet can be regarded as moving in an ellipse with the sun in one focus. The size and shape of the ellipse can be defined by the length of its axis and by the eccentricity; the plane in which the ellipse is situated is determined by the position of the line, called the line of nodes, in which it cuts a fixed plane, usually taken to be the ecliptic, and by the inclination of the two planes. When these four quantities are fixed, the ellipse may still turn about its focus in its own plane, but if the direction of the apse line is also fixed the ellipse is completely determined. If, further, the position of the planet in its ellipse at any one time is known, the motion is completely determined and its position at any other time can be calculated. There are thus six quantities known as elements which completely determine the motion of a planet not subject to perturbation.

When perturbations are taken into account, the path described by a planet in any one revolution is no longer an ellipse, though it differs very slightly from one; while in the case of the moon the deviations are a good deal greater. But if the motions of a planet at two widely different epochs are compared, though on each occasion the path described is very nearly an ellipse, the ellipses differ in some respects. For example, between the time of Ptolemy (A.D. 150) and that of Euler the direction of the apse line of the earth's orbit altered by about 5°, and some of the other elements also varied slightly. Hence in dealing with the motion of a planet through a long period of time it is convenient to introduce the idea of an elliptic path which is gradually changing its position and possibly also its size and shape. One consequence is that the actual path described in the course of a considerable number of revolutions is a curve no longer bearing much resemblance to an ellipse. If, for example, the apse line turns round uniformly while the other elements remain unchanged, the path described is like that shewn in the figure.

Euler extended this idea so as to represent any perturbation of a planet, whether experienced in the course of one revolution or in a longer time, by means of changes