150 ON THE ORBITS OF THE ASTEROIDS. 
The expressions for p and q of all the asteroids contain the common terms 
M sin y +h sin [1] + k sin [2] +.... + E; sin [7], 
M cos y + k, cos [1] + k cos [2] + .... + & cos [7]. 
The mean value of the first of these is about +.0180, and that of the second very 
small. Hence a common tendency exists among the nodes of the asteroids to be in 
90° of longitude. The second and third columns of the following table exhibit the- 
real and the probable distribution. 
and 
1 15 15 
2 "c 16 
3 13 13 
4 9 13 
The excess of the number in the third quadrant over that in the fourth proceeds from 
a cause similar to that which produces the excess in the perihelia, above referred to. 
In the general expressions for the eccentricity, inclination, longitude of perihelion, 
and longitude of node of Jupiter, the principal terms are 
Ae = + .0031 sin (0) + .0427 sin (1) — .0155 sin (2) 
ly = + .0031 cos (0) + .0427 cos (1) — .0155 cos (2). 
Pry = M sin y + .0012 sin [1] — .0063 sin [2] — .0015 sin [7] 
diy = M cos y + .0012 cos [1] — .0063 cos [2] — .0015 cos [7]. 
The comparison of the coefficients in these terms with the values of the correspond- 
ing e's and x's given in Tables I. and II. of $ 1, show that the corresponding quantities 
have the same signs, and that the different ratios of their magnitudes do not differ 
very materially from each other. The cause of this relation is, moreover, evident from 
an examination of the process by which the values of s and x were obtained. It 
follows from it that the general law of grouping of the nodes and perihelia of the 
asteroids may be expressed by saying that there is always a tendency in the perihelia 
of the asteroids to coincide in longitude with the perihelion of Jupiter, and in their 
nodes to coincide in longitude with the node of Jupiter. Sometimes, however, this 
tendency may be more than compensated by the circumstance of a number of the 
angles B+bt and C— bt having values nearly 180° different from the longitude of 
the perihelion, or longitude of the node of Jupiter. It will be most manifest in those 
asteroids which have small eccentricities and inclinations; thus the perihelion of Har- 
monia will very rarely be as much as 90° distant from that of Jupiter. For a similar 
reason the mutual inclination of the orbits of Euterpe and Jupiter will always be quite 
small, it being represented very nearly by V (p’—p)?+ (g — gy : 
