ON THE ORBITS OF THE ASTEROIDS. 147 
The computation of £, y, and £, with the data necessary thereto, are given in the 
preceding table. 
In this I have also included the quantities necessary for finding whether there 
exists any relation between the magnitudes of the asteroids, and the quantities 
é y, and č, or the velocities with which they were projected. The second 
column gives the value of the eccentricity used in the computation; and the third 
the difference between the actual mean distance of the asteroid and 2.56. Then follow 
the values of a! $, a*y, and a*t, as given by the formule (19). The seventh column 
contains the values of atv, which is proportional to the probable absolute velocity of 
projection of the asteroid, v being equal to y$} „>F Æ To express these velocities 
in ordinary astronomical units, they must be multiplied by EN If we take the 
earth's mean distance, and the solar day, as the units of space and time respectively, 
this factor will be .0107 ; and if for these units we take the English mile and the solar 
second, the factor will be 11.95. The eighth column gives the absolute velocity of 
projection of the asteroid, in miles per second. These numbers are not to be regarded 
as absolutely accurate, being separately subjected to possible errors of 4 or 5 in the 
last place. But from what precedes, it will be seen that the mean of a considerable 
number of them will be nearly exact. When £ is imaginary, its value has been sup- 
posed zero in computing v, and the latter has been put between parentheses. 
The column marked M gives a series of numbers proportional to the superficial 
area of the asteroids, as deduced from their brilliancy. The square roots of these 
numbers will therefore be proportional to the diameters of the asteroids. In obtaining 
the values of M, I have used the table of apparent magnitudes published by Mr. Pogson 
in the Monthly Notices of the Royal Astronomical Society, for January, 1859. 
The column headed m, gives a series of numbers showing the order of magnitude of 
the forty-eight asteroids contained in Mr. Pogson's table. 
It will be observed that a considerable number of the values of £ are imaginary. 
These values pertain to those orbits of which, with the tabular eccentricity, the peri- 
helion distance is greater, or the aphelion distance less, than 2.56. The real values of 
é are on the whole very nearly the same as those of 7; if we regard the imaginary 
values as zero, é will on the whole be sensibly less than ¢ Still their agreement is 
quite remarkable, and this favors Olbers's hypothesis, since, as before remarked, it is 
what might naturally be expected if this hypothesis were true. But the values of y 
are far less than those of E and £, which indicates that those fragments which were 
projected in the direction of the line of motion of the planet were thrown with much 
