e mass of the moon. If the moon is
in quadrature, the effect will be null. The coefficient of this
inequality is 90', and depends on the sun's distance from the moon. When
the moon is more than 90d from the sun, this correction is positive, and
when less than 90d from the sun, it is negative. If we call this second
correction C, and the moon's distance from her quadratures Q, we have
the value of C = +/-(90' x sin Q)/R.

[Illustration: Fig. 11]

This correction, however, does not affect the inclination of the axis of
the vortex, as will be understood by the subjoined figure. If the moon
is in opposition, the axis of the vortex will not pass through C, but
through C', and QQ' will be parallel to KK'. If the moon is in
conjunction, the axis will be still parallel to KK', as represented by
the dotted line qq'. The correction, therefore, for displacement, is
equal to the arc KQ or Kq, and the correct position of the vortex on the
surface of the earth at a given time will be at the points Q or q and Q'
or q', considering the earth as a sphere.

[Illustration: Fig. 12]

In the spherical triangle APV, P is the pole of the earth, V the pole of
the vortex, A the point of the earth's surface pierced by the radius
vector of the moon, AQ is the corrected arc, and PV is the obliquity of
the vortex. Now, as the axis of the vortex is parallel to the pole V,
and the earth's centre, and the line MA also passes through the earth's
centre, consequently AQV will all lie in the same great circle, and as
PV is known, and PA is equal to the complement of the moon's declination
at the time, and the right, ascensions of A and V give the angle P, we
have two sides and the included angle to find the rest, PQ being the
complement of the latitude sought.

We will now give an example of the application of these principles.

_Example._[10] Required the latitude of the central vortex at the time
of its meridian passage in longitude 88d 50' west, July 2d, 1853.

CENTRAL VORTEX ASCENDING.

Greenwich time of passage 2d. 3h. 1m.
Mean longitude of moon's node 78d 29'
True " " 79 32
Mean inclination of lunar orbit 5 9
True " " 5 13
Obliquity of ecliptic 23 27 32"
Mean inclination of vortex 2 45 0

Then in the spherical triangle PEV,

PE is equal 23d 27' 32"
EV " 7 58 0
E " 100 28 0