142 NINTH ANNUAL REPORT OF 



year is extremely small. The annual motion of the node of Mercury 

 is ten seconds ; that of Venus is twenty seconds ; Mars twenty-five 

 seconds, &c. The most rapid motion tbund in the nodes of any of the 

 asteroid orbits, as far as the computation has been made, is about fifty 

 seconds a year. If we suppose the nodal hues of all these orbits to 

 move steadily toward each other, it would require, in some of them, a 

 motion of fiftji seconds a year, continued for more than six thousand 

 years, to bring them to a coincidence. 



It must also be observed, that not only must the nodes of all the 

 asteroids coincide, but the distance of the planets from the sun must be 

 the same at that instant. Now, the distances of these bodies from the 

 sun, when at their nodes, differ by more than a hundred millions of 

 miles ; so that, to bring them all together, requires something more than 

 a change in the position of the nodes. We may bring about a coinci- 

 dence in the case of some of the asteroids, by supposing the longer 

 diameter of the elliptic orbit to change its position, without disturbing 

 the plane of the orbit. Such a change does really take place in the 

 case of every planetary orbit, but with none of the larger planets does 

 it exceed twenty seconds a year. In the case of one of the asteroid 

 orbits, this motion has been found to amount to seventy seconds a j^ear ; 

 but, even with this motion, it would require the lapse of five thousand 

 years to bring about an intersection in the case of many of the asteroid 

 orbits. When, now, it is remembered that, in order to give a common 

 point of intersection to these thirty-one orbits, all the nodal lines must 

 coincide, and, at the same instant, all the distances from the sun must 

 be equal to each other, we must be prepared to admit that such an oc- 

 currence could only have taken place myriads of years ago. 



The preceding difficulties, however, are small, in comparison with 

 another which remains to be stated. The orbit of Hygeia completely 

 encloses the orbit of Flora, and, indeed, several other orbits, and would 

 still enclose them, although the greater diameter of each of them were 

 revolved through an entire circumference — since the least distance of 

 Hygeia from the sun exceeds the greatest distance of Flora. The 

 same is true of ThemJs, as compared with Flora, and several other 

 orbits. The least distance of Hygeia from the sun exceeds the great- 

 est distance of Flora by more than tive7ity-Jive millifms of miles. In order 

 to render an intersection of these orbits possible, we must suppose a 

 great variation of the eccentricity. But the change of eccentricity of 

 the planetary orbits is exceedingly slow, and the present rate of increase 

 of the eccentricity of Vesta must be continued twenty- seven thousand 

 years, to render the aphelion distance of that planet equal to the peri- 

 helion distance of Hygeia. Moreover, the eccentricity of the orbit of 

 Vesta is now increasing, which implies that in past ages the interval be- 

 tween Vesta and Hygeia must have been greater than it is at pr'esent ; 

 whence the conclusion appears irresistible, that the orbits of Vesta and 

 Hygeia cannot have intersected for several myriads of years. When the 

 secular variations of the elements of each of the asteroids have been 

 computed, astronomers will be able to assign a limit of time, beyond 

 which the intersection of all the asteroid orbits must have occurred, if, 

 indeed, such an intersection ever took place. The discovery of many 

 of these bodies is so recent that, as yet, there has not been sufficient 



