82 On the Small Planets between Mars and Jupiter. 
gous difficulties, which we can eliminate only by considering the 
terms in which the longitude of these.elements does not enter, if 
such terms exist. Now the motion of the perihelion, whether of 
Mars or the Earth, actually contains a sensible term of this kind: 
this term depends only upon the mean distance of the asteroids 
from the sun and upon the excentricity of the disturbed planet; 
moreover it is essentially positive whatever may be that of the 
small planets whose action upon Mars and the Earth we are here 
considering, so that all these small masses combine by their action 
to impress direct motions upon the perihelia of the two principal 
planets here referred to. If then we suppose that the zone in 
which these small planets are found contains a very great num- 
ber of others like them, we should conclude that the whole group 
acts upon the perihelia very nearly as if they were collected into 
a single mass situated at a proper mean distance, and we should 
deduce therefrom the means of determining the total mass, or at 
least a limit which it cannot exceed. ft ; 
This subject however presents other difficulties. Besides the 
term upon which we have been reasoning there is a second in 
the expression of the motion of the perihelion, of the same mathe- 
matical order of magnitude as the first, but which depends upon 
the direction of the perihelia of the several disturbing masses: it 
is important to inquire whether it can modify the results fur 
nished by the first term. . 
situated in one half of the heavens, would be destroyed in this 
second term by the action of those masses whose perihelia are in 
the perihelia of twenty out of twenty-six being placed in one- 
half of the heavens, a result doubtless not of chance and seeming 
to indicate that the matter whose mass we are investigating is 
nearer the sun on the side of the summer solstice than of the 
winter. This circumstance must be taken into consideration, 
not for the purpose of introducing it as an essential condition into 
the solution of the problem, but on the contrary of arriving at @ 
result which shall be independent of it. ' (a 
This consideration will lead us not to make use of the motion 
of the earth’s perihelion although it is better known than that of 
Mars. The earth’s perihelion being in fact situated in that very 
portion of the heavens occupied by the perihelia of more than 
three-fourths of the asteroids, the second term which enters into 
the expression of its motion may become appreciable as com 
pared with the first and of the contrary sign: inasmuch as thes 
