730 
directed along the earth’s four-dimensional track and that the origin 
of space-axes falls along with the earth. Moreover, the original 
directions of these. space-axes at successive instants are to remain 
parallel to themselves in the general, or natural sense. If our axes 
of reference are chosen in this way, we may confidently expect the 
equations of motion to assume a particularly simple form: in fact, 
as a first approximation, when motions take place very near the 
origin (i.e. within a domain the two-dimensional cross-sections of 
which are small compared with the reciprocal of RigMANN’s measure 
of curvature) then this region may be considered to be homoloidal, 
that is, free particles are moving in straight lines under no force, 
and a top spinning round its axis of symmetry will keep its axis 
of rotation in a fixed direction relative to the axes of reference. As 
the latter are carried along the axis of time parallel to themselves, 
so it follows that the same is true for the axis of rotation. *) 
If we proceed to the second approximation, we find that free 
particles are subject first to forces which we know are the causes 
of the tides due to the sun’s action, and secondly, to forces depending 
on the velocity of the particle in a manner which in a certain 
respect resembles Corionis’ forces in a centrifugal field. The latter 
were called by Poincaré “forces centrifuges composées’’. Accordingly 
the new forces might be designed as compound tidal forces. 
In order to obtain the second approximation, it is necessary to 
specify our coördinates in greater detail. In every point-instant of 
the axis of time we draw all geodesic lines which are perpendicular 
to the time-track and we desire that these shall define space, three 
of them being chosen as the axes of space. For convenience sake 
the latter may be chosen perpendicular to each other. 
It will be seen that this space cannot coincide with space as 
defined by an observer who is at rest with the sun. The two spaces 
of reference intersect in a surface, which, in each point-instant of 
the earth’s helicoidal track contains the direction in the ecliptic 
perpendicular to the velocity and the direction perpendicular to the 
1) In much the same manner during the moon’s motion, as a first approximation, 
— apart from the sun’s perturbing forces, which arise in the second approxi” 
mation, — the plane of the orbit must keep its position unaltered relative to the 
falling axes of reference. This results in a motion of the nodes equal to the motion 
of these axes. De SirTeR, proceeding in a totally different manner, arrived at a 
nodal motion of 1”.91 per century, which is exactly the amount given above for 
the precession. (Monthly Notices R. A. S. 77, p. 172, 1916). A comparison with 
observation could only be made if the nodal motion, resulting from other causes 
and computed with Newron’s law of force, were known to one further decimal 
place than it is at present. (Dr Sitter, l.c.). 
