140 ON THE ORBITS OF THE ASTEROIDS. 
and, as the change is in each member of the same order of magnitude, the planets having 
large eccentricities can be treated without great error in nearly the same manner as those 
for which this element is small. We may, however, obtain a more simple inequality 
than (13), which will have the advantage of enabling us to ascertain how near any 
number of asteroids could ever have come to a common point of intersection. Suppose 
the conditions (14) to be fulfilled. "Then suppose any one of the angles (0), (1), (2), to 
vary, and let it differ by the quantity a from its first value. Then the second member 
of (13) will, in consequence of the change in the eccentricities of the planets, change 
its value by a quantity very nearly of the form u cos a, and the change in the first 
member will evidently be of the same form, and may be represented by v cosa. w and 
p are one half the changes which the first and second members of (13) respectively 
undergo by a change of 180° in the value of the angle. Hence the state of the orbits 
most favorable to (13) will be found either when « = 0, or when a = 180°, according 
to the relative magnitudes of u and w. The perihelion of the outer planet will then 
have the same longitude as the aphelion of the inner one; and since in the state supposed 
the eccentricities of the orbits are equal to (A +. +s +é... .) and(A FEF 
F. . . .), it follows that the condition of possibility of intersection will be ` 
a(lAtiata....En) >a(1—AFEF.).... Fe); (15) 
the corresponding ce having opposite signs in the two members, and being so taken 
as most to favor the condition. From Table I. it is evident that e, «,, and e, should be 
taken negatively (without regard to their signs in the Table); while & £j, &c. should 
be taken positively; and & is doubtful, but would generally have to be taken positively, 
e, Eu and e, must therefore be regarded as positive, and the other es as negative. 
From this, and from the preceding numerical expressions for the secular variations, ` 
we deduce the following system of values of the eccentricities, and consequent peri- 
helion and aphelion distances of certain of the asteroids which lie near the extreme 
limits of the zone, as being those most favorable to the intersection of their orbits in a 
common point. | 
OUTER ASTEROIDS. INNER ASTEROIDS. 
Eccentricities. Perihelion Distances. Eccentricities. . |Aphelion Distances. 
Hygea .1505 208 Vesta .0968 2.58 
Themis ` - .1639 9.63 Flora .1304 9.49 
Psyche .1190 9.58 Metis .1101 9.65 
Letitia ` ` .1094 9.49 Urania .0925 9.58 
Harmonia ` .0077 9.29 
From these values it follows that while the coefficients e, &, &c. and A have the 
