- ------- ) sin d.
2
k = a + ----------------------
R

The computation will be shorter, however, if we merely reduce the
inclination to the sine of the distance from the node for the first
correction of the arc AR, if we neglect the semi-monthly motion of the
axis; for this last correction diminishes the plus corrections, and the
first one increases it. If, therefore, one is neglected, it is better to
neglect the other also; especially as it might be deemed affectation to
notice trifling inequalities in the present state of the elements of the
question.

There is one inequality, however, which it will not do to neglect. This
arises from the displacement of the axis of the vortex.

DISPLACEMENT OF THE AXIS.

We have represented the axis of the terral vortex as continually passing
through the centre of gravity of the earth and moon. Now, by following
out the principles of the theory, we shall see that this cannot be the
case, except when the moon is in quadrature with the sun. To explain
this:

[Illustration: Fig. 10]

Let the curve passing through C represent a portion of the orbit of the
earth, and S the sun. From the principles laid down, the density of the
ethereal medium increases outward as the square roots of the distances
from the sun. Now, if we consider the circle whose centre is C to
represent the whole terral vortex, it must be that the medium composing
it varies also in density at different distances from the sun, and at
the same time is rotating round the centre. That half of the vortex
which is exterior to the orbit of the earth, being most dense, has
consequently most inertia, and if we conceive the centre of gravity of
the earth and moon to be in the orbit (as it must be) at C, there will
not be dynamical balance in the terral system, if the centre of the
vortex is also found at C. To preserve the equilibrium the centre of the
vortex will necessarily come nearer the sun, and thus be found between T
and C, T representing the earth, and [MOON] the moon, and C the centre of
gravity of the two bodies. If the moon is in opposition, the centre of
the vortex will fall between the centre of gravity and the centre of the
earth, and have the apparent effect of diminishing the mass of the moon.
If, on the other hand, the moon is in conjunction, the centre of the
vortex will fall between the centre of gravity and the moon, and have
the apparent effect of increasing th