18 
MR. LUBBOCK’S RESEARCHES 
tude of the perihelion, and the longitude of the epoch, contains a term which 
varies with the time, and hence the line of apsides and the line of nodes revolve 
continually in space. The stability of the system may therefore be inferred, 
which would not be the case if the eccentricity, the major axis, or the tangent 
of the inclination of the orbit to a fixed plane contained a term varying with 
the time, however slowly. 
The problem of the precession of the equinoxes admits of a similar solution ; 
of the six constants which determine the position of the revolving body, and 
the axis of instantaneous rotation at any moment, three have only periodic 
inequalities, while each of the other three has a term which varies with the 
time. From the manner in which these constants enter into the results, the 
equilibrium of the system may be inferred to be stable, as in the former case. 
Of the constants in the latter problem, the mean angular velocity of rotation 
n— 1 
d a = 
d e = 
2ca / fi\ 2 
~ n\aj i 1 + ne cos u + n (n + 1) e 2 cos' 2 u + &c.} du 
71—1 
— { 1 + n e°- + ne cos u + n ( n - + . ^ e 2 cos 2 0 _j_ & Ct j, d v 
n— 1 
2c / ^i\ 2 
— n \a/ {cos o + (n + l)ecos 2 u + (n + l) 2 e 2 cos 3 u -f &c.} (1 — e 4 ) du 
n— 1 
2 
2c / - fn + 1 / „(n+l)2\ , (»-H) 
= -^\a) {— e + (l + 3 4 “ ycoso + ecosZv 
( t \ _L J 
4- — 4 cos 3 v + &c. | (1 — e e ) d u 
neglecting the terms which are periodic 
w— l 
2 
v(f) + 
71 — ] 
The major axis decreases perpetually, the eccentricity diminishes perpetually until it reaches zero, 
while the perihelion retains the same mean position, and the longitude of the epoch the same mean 
value. I stated inadvertently in the former part of this paper, p. 340, that the variations of the 
eccentricity are all periodical. 
