ON THE ORBITS OF THE ASTEROIDS. 149 
§ 5. 
Of certain observed Relations among the Orbits of the Asteroids, which are the Result, in whole or in Y 
of known Causes. 
It has frequently been noticed by writers on the distribution of the asteroids, that 
the perihelia and nodes of these bodies are very unequally distributed in longitude. 
For about two thirds of the asteroids, these elements are found in the first semicircle 
of longitude. 
These inequalities of distribution proceed principally from the fact that some of 
the principal terms in the expressions for h, l, p, and q, given in $ 1, have common 
angles for all the asteroids, and that the coefficients of each of these angles have the 
same sign for the different asteroids. "Thus in the expressions (12) the terms 
e sin (0) + e, sin (1) + e, sin (2) + .... + e, sin (7), 
¿Cos (0) + s cos (1) + &cos (2) + .... + & cos (7), 
and 
are common to all the asteroids, the different «’s all having the same sign when they 
are of appreciable magnitude. The average value of the first of these expressions is 
about +.0011; and of the second, about .0314. These common terms, therefore, 
cause a tendency in the perihelia to be near the longitude of which the tangent is 
yy, or very nearly 0. About 33 of the 57 known asteroids have their perihelia 
within 90° of this point of longitude. This is but one more than the probable number 
which we should expect as the effect of the above-mentioned tendency. The perihelia | 
are distributed in the four quadrants as shown in the second column of the following 
table. The third column shows the probable number, taking into account the above- 
mentioned tendency. 
1 22 16 
2 15 13 
3 9 12 
4 11 16 
'The excess of the number in the first quadrant over that in the fourth, and of the 
number in the second quadrant over that in the third, proceeds from the unequal dis- 
tribution of the angles B around the circle; and this again is merely a chance tempo- 
rary accumulation of those angles in the first two quadrants, which, from their expres- 
sions in $ 1, will evidently not be permanent, but will wear away in the course of a 
few thousand years, and which did not exist a few thousand years ago. ` 
