159 ON THE ORBITS OF THE ASTEROIDS. 
the first place, if we consider the planet at the origin of its orbital motion, we see that, 
in order that its orbit may be very nearly circular, two independent improbable con- 
ditions must be fulfilled ; — firstly, that the direction of its motion shall be very nearly 
at right angles to the line passing through the planet and the sun; secondly, that its 
velocity should be very nearly equal to T r representing the distance of the planet 
from the sun. If we regard the probabilities of these separate circumstances as small 
quantities of the first order, the probability of their concurrence will be a small quantity 
of the second order. The probability that the eccentricity does not exceed a small 
quantity, c, will therefore be proportional to d so long as ø does not exceed a certain 
narrow limit. 
The same reasoning can be applied to the inclinations. In order that the inclination 
of the orbit of a planet to a plane taken at random shall be very small, it is requisite 
both that the planet should be very near this plane, and that the line of direction of its 
motion should be very nearly in this plane. 
To show the same result in a general form, we observe that h and / represent the 
. negatives of the co-ordinates of the centre of the ellipse in which the planet is moving, 
when the mean distance of the planet is taken for unity. If now we project the positions 
of these points on the plane of the ecliptic, we might expect to find those near the sun 
distributed nearly at random. If we draw a circle with a small radius, o, another with 
a radius 29, &c., around the sun as a centre, and if the centres of the orbits are equally 
distributed, the space between the first and second circles will contain three times as 
many centres as the inner one, the space between the second and third five times as 
many, and so on. It will be perceived that there is really a deficiency of small eccen- 
tricities, and a superabundance of small inclinations, though neither irregularity is 
greater than what might result from chance deviations in distribution. 
It has been suggested by an acute astronomer that the excess of small inclinations 
proceeds from the fact that observers generally look for asteroids very near the plane of 
the ecliptic. Considerable weight is given to this supposition by the circumstance that 
most of the asteroids-have been near their node at the time of their discovery. 
It seems highly probable, from this circumstance, that the mean inclination of the 
whole number of asteroids, known and unknown, is very much greater than that of 
the known ones. If so, the fact furnishes an additional argument against the hypoth- 
esis of explosion, since £ must then be much greater than ¿. 
