Mathematical tracts on the lunar and planetary theories, the figure of the earth, precession and nutation, the calculus of variations, and the undulatory theory of optics.

SECULAR VARIATIONS OF ELEMENTS. 119
whencede, m' d C(~
whence..= —na —.
dt,u da
If the orbit of m be exterior to that of m', or if a be >a',
dC(o)
it appears frorn the expansion of (121) that d is negative,
and therefore ' is positive. Let it = B: then e, = e + Bt:
dt
hence the mean longitude or nt + e, = (n + B).t +e. This
shews that the mean motion is greater than that in an undisturbed orbit with the same axis major. {We have in (91)
arrived at a result equivalent to this.} If a were less than
a dC(O)
a:, C(~) would be expanded by powers of -, would
a da
be positive, and the mean motion would be less than in
an undisturbed orbit.
142. It is almost unnecessary to remark that the disturbing effect of several planets will be the sum of their separate
disturbances. This always holds when the equations for the
perturbations are linear, as in the planetary theory.
143. We have stated that though we ought in strictness
to substitute in every term of R the variable values of the
elements, yet it is generally sufficient to use constant values.
But this is hardly accurate enough in the calculation of some
equations of long period, where e. and ar alter considerably in
the period. In general however it will be sufficient to sub,
stitute for them e + Et and r + Ilt, {where E and In are the
values of dt and d- found in (141)} in the values of
dR dR dR
de' d'r' de
and then proceed with the integration.
144. PRO,. 58. To investigate the secular variations of
the eccentricity and longitude of perihelion.
We have now sufficiently explained the method of calculating those variations of the elements whose period is