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.

86 PLANETARY THEORY.
most materially changed, yet their value may not sensibly
change from one year to another, so that a single revolution (so far as these long inequalities alone are concerned)
may be very exactly represented by the ellipse whose elements are the values of the elements corresponding to any
instant of that revolution. For the same reasons it is particularly well adapted to the discovery of those changes
which continue constantly in the same direction, as for instance, the motion of the perihelion.
107. PROI'. 40. To investigate the alteration of the
semi-major axis in a disturbed orbit.
We shall, as before, put a, e, my, for the invariable
elements in an ellipse nearly coinciding with the curve described, and a, e,, r,, for the variable elements which will
represent the place and velocity of the planet in the curve
described, provided the calculation be made for any instant
as if they were invariable. Now upon examining the process above, it will be seen that the place and velocity or
first differential coejlcient in the real curve are the same
as in this instantaneous ellipse: but there is no such condition respecting the second differential coefficient. We
must then use the equations which have been once integrated. Now in an undisturbed elliptic orbit this equation
is easily found, by the same process as in (79) and (80),
c d=r,2-ir d 2+
C = dt) ' ddt t)
If we make this
( dr1)i T r dOd + 2 2-L
du= { 'o +2 h-2
as in (10) and (11), and put for u and h the values there
given, the semi-major axis being a,, we find C= —
Thus in ad m n in or f s
Thus in undisturbed motion in our fictitious orbit,