198 ON THE ORBITS OF THE ASTEROIDS. 
time. If, now, we suppose the radius vector pi move around the end of M 
1 
^ 
> ‘ : D - : 
with a uniform angular motion equal to E, Pk to move around with an angular 
2 e 
motion of k,, and so on to K; the end of K will continually be on the point in which 
the pole of the orbit of the asteroid intersects the surface of the sphere. 
M and y are the same for all the planets and asteroids; the end of M may therefore ` 
be taken as an origin around which the poles of the orbits of all the planets and aste- 
roids move. This point is evidently the mean position of the pole of each separate 
planet and asteroid during all time,* and the plane of which it is the pole may be 
regarded as the mean position of the plane of the orbit of each heavenly body during 
all time. This plane is the invariable plane of maximum areas; as may be seen from 
the fact that the constant M vanishes when we take that plane as the plane of reference.T 
This plane ought, also, to be the probable mean position of the orbits of the several 
asteroids at any one time, in so far as the positions of the separate orbits are inde- 
pendent of each other, and to the same degree, the nodes on this plane ought to 
be distributed at random. We shall return to this subject when considering the 
distribution of the nodes and perihelia in longitude. : 
In the above investigations and constructions, quantities of the third order with 
respect to the eccentricities and inclinations have been neglected, and we have there- 
fore made no distinction between the distance from the origin of the poles of the 
orbits on the surface of the sphere, and of the points in which these poles intersect 
the tangent plane, or the secant plane which passes through the point, and is parallel 
to the tangent plane; or between i, sin 7, and tang i. 
In the above construction, the distance of the final point, or pole of the orbit 
of the asteroid, from the pole of the ecliptic, will be equal to the inclination, and 
its longitude increased by 90° will be equal to the longitude of the node. If, then, 
en I * 
one of the radii vectores bb 0 K is longer than the sum of all the others, it 
is evident that the amount by which it exceeds that sum will be an inferior limit of the 
inclination, and that the mean motion of the node will be equal to the coefficient of + 
in the angle which corresponds to the longest radius vector. 
Let us now apply the formule given above to the numerical computation of the 
elements of the asteroids in terms of the time. Many of the required quantities being 
* In other words, if we mark on the plane tangent to the sphere, and parallel to the invariable plane, the 
points in which the pole of the asteroid orbit intersects it at equidistant intervals of time, to infinity, the point 
of tangency will be the centre of gravity of all these points. 
t Méc. Céleste, Liv. II. No. 62. 
