JANITABY 20, 1911] 



SCIENCE 



91 



field for speculation which I shall not at- 

 tempt to enter this afternoon. 



I have hitherto spoken of asteroids which 

 belong to the main group. Others are scat- 

 tered either singly or in small groups at 

 quite different distances from the sun. 

 Bros, whose orbit lies within that of Mars, 

 is so well known owing to its value for the 

 determination of the parallax of the sun, 

 that it needs only a passing mention. A 

 small group of five asteroids just inside the 

 mean radius at which the period is two 

 thirds that of Jupiter, is of interest as an 

 illustration of the features of those aster- 

 oids the large perturbations of which de- 

 pend on the first power of the eccentricity. 

 As at the place where the ratio is 2:1, the 

 planets are on the inner side of the gap. It 

 is, of course, not possible to draw any con- 

 clusions from so small a number. 



But the most remarkable illustrations of 

 the problem of three bodies are four small 

 asteroids which have been found within 

 the last five years. Before that time the 

 period and therefore the mean distance 

 from the sun of every known asteroid were 

 less than those of Jupiter. In February, 

 1906, Wolf observed a body whose velocity 

 appeared to be unusually small and, in the 

 course of a few observations, it was seen 

 that its orbit could not be very distant 

 from that of Jupiter. Charlier suggested 

 that it might probably be an illustration of 

 one of the two eases in which the problem 

 of three bodies can be solved without ap- 

 proximation. Long ago Laplace proved 

 that if three bodies be placed at the corners 

 of an equilateral triangle and started re- 

 volving round their common center of mass 

 with a properly adjusted angular velocity, 

 they would continue to do so indefinitely. 

 The triangle remains equilateral and each 

 body describes an ellipse with the center of 

 mass as a common focus. Further, it has 



been proved that, provided the masses 

 satisfy certain relations, the motions are 

 stable, that is, small oscillations about the 

 triangular positions would not involve the 

 ultimate loss of the type of motion ; the tri- 

 angle would always remain nearly equi- 

 lateral. As the mass of the asteroid is so 

 small that it may be neglected, the condi- 

 tion for the stability of small deviations is 

 that one mass shall be at least twenty-five 

 times the other; since the sun is over 1,000 

 times as heavy as Jupiter the condition is 

 well satisfied. There seems to be no doubt 

 that the theory is a correct explanation of 

 the facts. In the intervening j^ears three 

 more asteroids with similar orbits have 

 been discovered. Of the four now known, 

 three are near the point 60° in advance of 

 Jupiter, and one is near the point 60° be- 

 hind. The mean distances of all four from 

 the sun differ but little from that of 

 Jupiter. 



If we examine the orbits of these aster- 

 oids on the hypothesis mentioned, namely, 

 that they are merely deviations from the 

 triangular solutions, we have to admit at 

 onee that the deviations can not be con- 

 sidered very small. One of them may move 

 as far as 17° away from the mean position 

 as seen from the sun, and a question nat- 

 urally arises as to the permanent stability 

 of such large deviations. The periods of 

 the oscillations must be very long, not less 

 than 150 years, so that we can not hope to 

 settle the question by direct observation. 



These oscillations may be regarded as 

 librational. Bodies well beyond the orbit 

 of Jupiter will revolve around the sun 

 more slowly than that planet, those inside 

 more quickly, and between the two we get 

 a region in which the mean period may be 

 the same. There may then be librations 

 about this mean period which carry the 

 body first inside and then outside the orbit 

 of Jupiter. Referring back to what has 



