736 
Resuming, we can say that as a first approximation the equations 
of motion for free particles in the geodesic falling system are just 
the same as those in classical dynamics under no forces. When we 
have mutual forces between the particles, their effects on the motions 
will be quite the same as predicted by classical dynamics. In parti- 
cular, a spinning top will keep the direction of its axis of rotation 
unaltered relative to the axes of reference, i.e. our geodesic falling 
coordinates. Hence when referred to the original coordinates, the 
spinning top will for its axis of rotation show whatever precession 
the geodesical falling axes might exhibit. 
The same must be said for the plane of the orbit of a particle, 
moving under a central force. 
If the tidal forces are considered, their effect in changing the 
direction of the axis of rotation relative to the falling coordinates 
would be zero if the earth were of spherical shape. If not, the 
precession caused by them is to be taken in reference to the falling 
axes, and the precession of the latter will be superposed on the 
precession due to the tidal forces. 
The common tidal forces are but part of the second approximation. 
The remaining part is a compound tidal force at right angles and 
proportional to the velocity, proportional to the distance from the 
centre and, like the Coriomisian forces, may be determined as a (three- 
dimensional) vectorial product of the velocity into a vector which, 
by means of certain components of the KimMannian bivector-tensor 
of curvature, is a linear function of the radius vector from the 
centre. For the present we shall leave these forces aside, and turn 
to the question of how much the amount of the precesssion of the 
falling axes may be. 
The precession of the geodesic falling axes in the case of ‘a 
planet moving in a circular orbit. 
AS we pointed out already, a complication in finding the precession 
of the falling axes arises from the fact that the space of the falling 
axes makes some angle with space as defined by an observer who 
has his coordinates fixed to the sun. These spaces intersect in a 
plane perpendicular to the velocity. By confining ourselves to circular 
orbits, matters present themselves much less complicated. 
In each point-instant of the helicoidal track of the planet we draw 
four local axes: one coinciding with the direction of the track; a 
second in the direction away from the sun along a radius vector; 
a third perpendicular to the ecliptic; and the last one with a time 
