Page:Encyclopædia Britannica, Ninth Edition, v. 2.djvu/846

780 periods in which the planets pursue their different paths. He did not at first try to connect the periods and the dis tances by any direct numerical relation, probably because he had recognised in the second law the probable existence of geometrical relations. But after many years of inquiry he arrived at the conclusion, that probably the required law connected the powers of the numbers representing the periods and the distances. It affords a strange evidence of the ponderous nature of Kepler s movements, that after this idea had occurred to him, ten weeks, instead of some ten minutes, elapsed before he had verified it. The law con necting the periods and distances Kepler s third law is this— 3. The square of the numbers representing the periodic times of the planets vary as the cubes of the numbers representing their mean distances. Or thus, If D, d be the distances of any two planets, and P, p their respective periods, then

D 3 (I)- From this law we can deduce a convenient relation between the velocities in circular orbits (a relation holding approximately for the mean velocities in orbits nearly circular like those of the planets). Let V, v be the velo cities in the orbits of planets whose distances are D, d respectively; then, obviously,

V&#61;c &amp;gt; and v c- &amp;gt; where c is some constant. P p V&#61;D p_&#61;D v d P d D* that is, the velocities in circular orbits vary inversely as the square roots of the distances. And also,

V P* p _p&amp;gt; ,, -&#61;-?-- ;a w V 2&amp;gt; : that is, the velocities in circular orbits vary inversely as the cube roots of the periods. It may be well also to notice the following relation between the angular velocities in such orbits. Let these, for the respective planets just dealt with, be f2 and o&amp;gt;. Then,

-P.- S!L. u P D* n i) (It is manifest that ~ p &amp;gt; because the periods must be inversely proportional to the angular velocities with which they are uniformly described.) The three laws of Kepler are approximately true for bodies circling around the same centre. They do not apply to bodies circling around different centres. For instance, the moon s distance and period could not be used for p and d in (1). Nor could the distance and period of any one of the satellites of Jupiter pair, according to Kepler s third law, with the distance and period either of our moon or of any planet ; but the motions of the satellites were found to accord with the law when com pared together.

It was probably the recognition of this fact which first put astronomers on the track of the theory that the law depends on some force residing in the centres round which different bodies move. Newton certainly had given atten tion to this influence before he dealt with the moon s attraction earthwards as a case of the action of terrestrial gravity. But, be that as it may, it is certain that so soon as the action of the earth s attraction on the moon had been demonstrated by him he extended the law of gravita tion to all cases of motion around a central orb. It then became clear that the laws of Kepler are consequences of the general law of gravitation the law, viz., that Every particle of matter in the universe attracts every other particle with a force varying directly as the masses, and inversely as the square of the distances. The proof of the law of gravitation divides itself into three distinct parts:— First, The proof that the force acting on the moon is equal to the force of terrestrial gravity, reduced as the inverse squares of the distances of the moon and of a point on the earth s surface, from the centre of the earth. Secondly, The proof that a system of bodies circling around a central body like the sun, attracting them with a force inversely proportionate to their respective distances, would obey the laws of Kepler, or some modification of those laws, giving results according with the motions actually observed. Thirdly, The proof that the mutual attractions of the several members of any system, and the attractions of members of one system on bodies belonging to another system (as, for instance, of the sun upon the moon regarded as a dependent of the earth), would result in such pertur bations from the paths due to the attractions of the central body as are observed actually to take place. Neither the second nor the third of these arguments can be given here, though certain simple relations involved in them, as also certain consequences, will be mentioned. The first part of the proof is altogether simple. The moon is, roughly, at a distance from the earth s centre equal to 60 radii of the earth, and therefore the earth s moving force is less on her than on a body at the earth s surface as 1 to 3600. Now, regarding the moon s orbit as a circle, it is easily shown that, if at any moment the earth s attraction ceased to act, so that for the next second the moon moved on a tangent to her present course, her distance from the earth s centre at the end of that second would be rather more than j^g-th of an inch greater than at the beginning of the second. It follows that her fall towards the earth in a second on account of the earth s attraction amounts to rather less than ~ ff of an inch. But the fall of a body near the earth s siirface is about IGj^- feet, or nearly 193 inches per second, or nearly 193 x 19 times greater than the fall of the moon towards the earth per second ; that is, about 3600 times greater. In other words, the moon is attracted towards the earth precisely as she would be if the force of gravity acting on bodies near her surface ruled her also, the law of variation of the force with distance being that of the inverse squares. The second part of the proof is simple for cases of circular motion of various bodies around a common centre. The law of the eqxiable description of areas, indeed, is true for a body moving around a centre attracting according to any law, since it simply implies that there is no force per pendicular to the radius vector. It is easily seen that the increase of the area during any exceedingly short interval of time depends solely on the distance attained by the moving body during that interval from the line represent ing the position of the radius vector at the beginning of the interval, this distance being measured in a perpendicular direction ; for the area of a triangle is measured by the base x perpendicular. Accordingly, if, during the short interval of time, there is no force tending to increase or diminish the perpendicular distance of the moving body from the original radius vector, as compared with the dis tance which would have been attained had no force at all acted, the area described will be the same as though no force had acted. But if no force acted, the body would move uniformly in a straight line, and the radius vector would sweep out equal areas, because triangles having the same vertex and their bases in one straight line have areas 