Page:Encyclopædia Britannica, Ninth Edition, v. 2.djvu/847

proportional to their bases. Hence, under a force not tending to change at any instant the rate of the moving body s perpendicular departure from the radius vector, the areas swept over in equal times will be equal. But a central force acts always in the direction of the radius vector, and is therefore a force of the kind supposed. Accordingly, a body travelling round an attracting centre will move so that the radius vector sweeps over equal areas

in equal times.

Suppose next two bodies describing circles with uniform motion in periods P, p, at distances D and d from a centre of attraction whose force varies inversely as (distance) 2 . It is required to determine the relation between P, p, D, and d. The velocities of the bodies are c -=- and c - > where P p c is some constant. Represent these velocities by V and v respectively for convenience ; also call the forces acting on the bodies respectively F and/, which we know to be proportional to^ and -^ Clearly, if a body moving in any direction receives a very slight impulse in a direction at right angles to its motion, its direction of motion will be changed through an angle proportional directly to the impulse and inversely to the velocity of the body. So that if we regard the attractive force on the planets as acting by a succession of small impulses, the momentary F variation of direction is proportional to ^ > and the time, therefore, of completing any given change of direction is y proportional to -^ Now, in times P, p respectively, the two planets have their direction changed through four right angles. Hence

V v D I : P jr=j - "p That is, P 2 : p 2 : : D 3 : d 3 , which is Kepler s third law.

Let us now compare the periods P and p of two bodies, moving at distances D and d, around unequal centres of force, which exert attractions A and a respectively at equal distances, so that instead of the forces F and / actually exerted on the two bodies being proportional to and -, they are proportional to =-^ and -^-respectively.

Then the above proportion becomes

D 2 : - P That is,

P D 3 A" A cP_ a D 3 Prt J- 7 - oc A corresponding to a law which may be thus expressed: The squares of the periods about different centres of force vary directly as the cubes of the distances, and inversely as the attractions of the centres of force at a unit of distance.

This relation which is true for elliptic orbits, is general for all systems, and gives the means of comparing the masses of different systems. But it is necessary to observe a modification which Kepler s third law and this extension nf it have to undergo to make them strictly true (as re gards, at least, the unperturbed motions of the planets). The masses of the planets, though very small, yet bear definite relations to the sun, and instead of considering each planet as swayed by the. sun s mass, we must regard each as swayed by the sum of its own mass and the sun s, supposed to be gathered at the sun s centre. Thus we must regard the planets as revolving around centres of different attractive energy; Jupiter round a centre equal in mass to Jupiter and the sun ; Saturn round a centre equal in mass to Saturn and the sun, &c. Instead, then, of . (mean distance)^ . . the ratio -- -, -^- being constant for the solar sys tem, we find that this ratio for any given planet is propor tional to the sun s mass added to that planet s. Extending the law to bodies travelling around different centres, it runs as follows:—

If a body of mass m revolves round a centre of mass M in time P, and at a mean distance D, and another body of mass m revolves round another centre of mass M in time P , and at a mean distance D , then

D 3 D 3 P 2 (M +roO

This law enables us at once to compare the sums of the masses when we know the mean distances and periods. For it may be written

M + m D 3 Pj M + m = D a P2 Also, where m and in are both small, compared with M and M respectively, the law becomes simplified into

M_ D 3 ^ P^ 2 M ~ D 3 P 2

These laws suffice to enable us to deduce from the observed periods of the planets their true mean distances, velocities, &c., and from the observed period of the satellite of any planet, the ratio of the planet s mass to the sun s. The eccentricities of the planetary orbits are partly deduced from observation, and partly from the law of the equable description of areas. The inclinations of the orbits, and, of course, all elements relating to the planets own globes, their dimensions, compression, inclination, rotation, and so on, are obtained by telescopic observation and measurement.

The following tables of elements are brought into one place for convenience of reference, and include many items of information referred to in the chapters on the several planets. The illustrative Plates numbered XXVII. and XXVIII. should be studied in combination with the table of planetary elements.

We may conveniently add here to the laws of planetary motion presented above the two following theorems (enunciated by Lagrange, but first established by Laplace).

First, If the mass of each planet be multiplied by the square of the eccentricity, and this product by the square root of the mean distance, the sum of the products thus formed will be invariable.

Secondly, If the mass of each planet be multiplied by the square of the tangent of the orbit s inclination to a, fixed plane, and this product by the square root of the mean dis tance, the sinn of the products tlius formed will be invariable.

These laws ensure the stability of the system. It is true that the whole eccentricity, if it could by any possibility fall on any one planet (except Jupiter or Saturn), would cause the planet s orbit to interfere with the orbits of other planets, or even, in the case of a small planet, to intersect the sun s globe. Yet the interchange of eccen tricities can never lead to this result. In fact, the sum of the products (mass) x (eccentricity ) 2 x (mean distance)* for Jupiter and Saturn will always largely exceed the sum of all such products for the remaining planets.

The fixed plane from which the inclination is to be measured, in the second law, is nearly identical with tlio plane of Jupiter s orbit. Its inclination to the ecliptic is about 1 35 , and the longitude of the node (in 1850) was about 104 21 . Comparing these values with the cor responding elements on next page, we see that the inclina tion of the planetary orbits would have a smaller mean value if estimated from the invariable plane than they had as estimated from the plane of the ecliptic.