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from the light and life of the planetary scheme. From Mars and Jupiter, and perchance from Saturn, Uranus, and Neptune, the unhappy career of the Earth might be traced for many a long year (though years—at least terrestrial ones-would then be no more). But long before the Earth crossed the confines of those distant regions along which the outer planets pursue their career, all the higher forms of life would have vanished from her surface. She would still rotate; day and night would still succeed each other on her surface; but the orderly sequence of the seasons would be replaced by the continual diminution of solar light and heat, until a cold more intense than that of the bitterest Arctic winter would bind the world in everlasting frost.

A similar fate would befall us if the Sun's mass were suddenly reduced in the proportion of about 1,000 to 1,414; the only difference being that in this case we should have companions in our troubles, for Mars and Venus and Mercury would all forthwith start on parabolic paths, carrying them away to infinite distances from the Sun. Nor would the larger planets escape. From their distant orbits they would rush off into outer space, carrying their systems of satellites with them; so that if I have been right in regarding these orbs as acting the part of secondary suns to their satellites, the latter would be less unfortunate in their fate than the four minor planets, for these would have no sun at all, while the former would still enjoy such heat and light as their ruling centres could supply to them. It is not without a purpose that I have thus dwelt

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on the general result of a sudden diminution of the Sun's mass. The consideration that all the planets would thus at once be freed from their allegiance if the Sun's mass were reduced, leads us to the consideration that each planet's velocity need but be increased in the proportion of about 1,414 to 1,000, to lead to a

similar result. And thus we see that the Sun's influence at the distances of the successive planets is limited to the control of bodies moving with a velocity bearing such a relation to the velocity of the respective planets. So that we have only to draw up a table of the distances and mean velocities of the planets, and to increase the latter quantities in the proportion of about 1,414 to 1,000, in order to have a representation of the gradual diminution of the Sun's influence at greater and greater distances. The table runs thus:

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We see from this table that if the three outermost planets could only have imparted to them the velocity of the minor planets, they would be freed from their allegiance to the Sun, and pass away on hyperbolie orbits. But with the exception of Uranus there is no

planet which would be thus freed (absolutely) if it had imparted to it the velocity of the next inner one.*

But the above table has only been presented by way of introducing a more general law. What the table teaches us respecting special distances we can determine for all distances from the Sun, by a simple application of Kepler's third law and its results.

Thus, suppose we wish to determine the maximum velocity which the Sun can control at a distance half that of the planet Mercury. Then the law that the

* It is a rather singular circumstance that the maximum velocity which the Sun can control at the distance of any one of the four outer planets should correspond so closely as it does with the actual velocity of the planet whose orbit lies next within. We see that if Neptune could have the velocity of Uranus, he would be almost wholly freed from his allegiance; Uranus would be just freed if he had the velocity of Saturn; Saturn would be almost wholly freed if he had the velocity of Jupiter; while Jupiter would be almost wholly freed if he had the velocity of those asteroids which travel at a mean distance. And there is a tendency, though less marked, to the same relation among the remaining planets. Remembering that the velocity a planet would require for freedom is that with which a body approaching on a parabolic orbit from an infinite distance would pass the mean distance of that planet, we have throughout the solar system a tendency (very marked among the outer members) to this remarkable relation, that the velocity with which a body approaching from infinity would cross the orbit of any planet should be the same as the actual velocity of the next inner planet. It need hardly be said perhaps that this relation directly results in the law to which Bode's law approximates for the outer planets. Thus, if the outermost planet had a distance D and a velocity v, the next minor planet would have a velocity v2 corresponding to a mean distance, and so on. But in the law of the duplication of the distances outwards there is no direct physical significance, whereas it is possible to conceive that the law as presented above —that is, regarded as associated with the velocities—may be associated also with the processes by which the solar system has reached its present condition.

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cubes of the distances are as the squares of the periodic times shows us that the period of a planet at such a distance would be to Mercury's as 1 to the square root of 8 (or twice the square root of 2). Since, then, the actual circuit of such a planet would be half that of Mercury, its velocity would exceed Mercury's in the proportion which the square root of 2 bears to 1,* or about 1,414 to 1,000. This would correspond to a velocity of 41.4 miles per second; and the greatest velocity the Sun could control at this distance would be obtained by increasing this velocity of 41.4 miles per second in the proportion of 1,414 to 1,000. It would therefore be 59.6. This shows how we can measure the Sun's controlling energy for any distance. But it also establishes a very important general relation. It appears that when we halve the distance, we have, in order to determine the velocity which the Sun can control, to increase the velocity at the greater distance in the proportion of about 1,414 to 1,000. And therefore when we take one-fourth of the distance, we must increase the velocity twice in this proportion. But this amounts to doubling the velocity, since this proportion is that of the square root of 2 to unity. Hence we have this general rule, that the velocity which the Sun can control is doubled when the distance

*This illustrates the general law that if two planets have mean distances d and d' respectively, and mean velocities v and respectively,

then

v: v' :: d''; d.

Clearly this is so, since the periodic times are as d to ď1, and the velocities, therefore, as d÷d to d'÷d.

is reduced to one-fourth, and we can see at once how enormously the velocity must increase in the Sun's immediate neighbourhood.*

Thus, at a distance of 8,848,000 miles (one-fourth that of Mercury), the velocity the Sun can control, so as to compel a body to move in a closed orbit around him, is 82.8 miles per second; at a distance of 2,212,000 miles it is 165.6 miles per second; at a distance of 553,000 miles it is 331-2 miles per second. But this brings us very close to the Sun's surface-since it is from his centre all our distances are measured-and his radius is about 426,450 miles. his surface—that is, the velocity which he could just control so as to compel a body to travel in a closed orbit just touching his surface-is easily obtained from the formula given in the preceding note. It is no less than 378.9 miles per second; and this is the least velocity with which a body must be projected from the Sun in order that it may never return to his globe again.

The actual velocity at

* It need hardly be said that this result might have been obtained directly from a consideration of the law according to which gravity diminishes with distance. But apart from the fact that the mere dry reasoning by which the result would have been established would have had little interest to the general reader, the particular path which I have selected to follow has the advantage of introducing a number of independent relations, and of showing how the various matters dealt with bear upon each other and upon the general subject of the chapter.

The general law connecting distance with the velocity which the Sun can control is as follows:-Let d represent the Earth's mean distance; v her velocity at that distance in miles per second; D any other distance. Then the velocity which the Sun can control (so as to compel a body to travel in a closed orbit round him) at a distance D is

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