ON THE ORBITS OF THE ASTEROIDS. 151 
The fact that the orbit of every asteroid, or nearly every one, is interlinked with the 
orbits of one or more other asteroids, so that if they were material we should by re- 
moving one carry off all the others with it, has sometimes been adduced as indicating 
a connection of some sort between these bodies. Let us examine the conditions of 
such interlinking. Suppose that one orbit of any pair is revolved around its node on 
the other orbit, as an axis, till the planes of the two orbits coincide. If their elements 
: fulfil the condition (13), 
(kU — kl)? + (kh! — Kh} > (e — ky, 
the orbits will then intersect in two points. If, on this supposition, they do not inter- 
sect, it is evident that they cannot interlink; hence the preceding condition is one 
which must be fulfilled to render it possible for them to interlink. It is also necessary 
that, in the position supposed, these points should fall on opposite sides of the line of 
nodes. Now, in view of the small differences of mean distances, and considerable 
eccentricities of the asteroids, it cannot be regarded as at all singular that a large num- 
ber of pairs should fulfil these conditions. As the orbits pass through their secular 
variations, some pairs which now interlink will cease to do so, and others which now 
do not interlink will do so. A change of this kind in some pair of orbits my be ex- 
pected to occur in nearly every century. Hence the fact of interlinking does not 
indicate any relation among the asteroids other than their being found together in a 
continuous zone; and can throw no light whatever on the question of their origin. 
In looking over a table of the elements of the asteroids, it is quite noticeable that the 
inclinations have a much wider range than the eccentricities. "Thus, while there is but 
a single asteroid the eccentricity of which is less than .06, there are ten or twelve whose 
inclination is smaller than this quantity. Again, several of the inclinations considerably 
exceed the superior limit of the eccentricities. This may be seen by the following 
table, which exhibits the distribution in magnitude of the eccentricities and inclinations 
used in the preceding section: — 
es vs 
t From .00 to .05 1 8 
.05 to .10 7 15 
.10 to .15 90 16 
.15 to .20 10 7 
.90 to .25 14 4 
.25 to .30 4 3 
Above .30 1 4 
It will also be observed, that there is a relative deficiency in the number of the above- 
mentioned elements having small values. The latter fact is easily accounted for. In 
