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A point-track is a locus of event-particles. It is defined by reference to one particular time-system, α say. Corresponding to any other time-system these will be a different group of point-tracks. Every event-particle will lie on one and only one point-track of the group belonging to any one time-system.
Thus each event-particle has position in this sense. The simplest mode of expressing the position in nature of an event-particle is by first fixing on any definite time-system. Call it α. There will be one moment of the temporal series of α which covers the given event-particle.
This is the theory of relative motion; the common matrix is the bond which connects the motion of β in space α with the motions of α in space β. Motion is essentially a relation between some object of nature and the one timeless space of a time-system. An instantaneous space is static, being related to the static nature at an instant.
Also every duration which is part of a given duration intersects the stations of the given duration in loci which are its own stations. By means of these properties we can utilise the overlappings of the durations of one family that is, of one time-system to prolong stations indefinitely backwards and forwards. Such a prolonged station will be called a point-track.
Namely, a straight line in the space of time-system β is the locus of those points of β which all intersect some one point-track which is a point in the space of some other time-system. Thus each point in the space of a time-system α is associated with one and only one straight line of the space of any other time-system β.
And that something which is measured by a particular measure-system may have a special relation to the phenomenon whose law is being formulated. For example the gravitational field due to a material object at rest in a certain time-system may be expected to exhibit in its formulation particular reference to spatial and temporal quantities of that time-system.
The explanation is not given here . It is sufficient now merely to mention the sources from which the whole of geometry receives its physical explanation. Cf. Principles of Natural Knowledge, and previous chapters of the present work. The correlation of the various momentary spaces of one time-system is achieved by the relation of cogredience.
The observed motion of an extended object is the relation of its various situations to the stratification of nature expressed by the time-system fundamental to the observation. This motion expresses a real relation of the object to the rest of nature. The quantitative expression of this relation will vary according to the time-system selected for its expression.
According to the theory of these lectures the axes to which motion is to be referred are axes at rest in the space of some time-system. For example, consider the space of a time-system α. There are sets of axes at rest in the space of α. These are suitable dynamical axes. Also a set of axes in this space which is moving with uniform velocity without rotation is another suitable set.
For consider the space of a moment M. Let α be the name of a time-system to which M does not belong. Let A₁, A₂, A₃ etc. be moments of α in the order of their occurrences. Then A₁, A₂, A₃, etc. intersect M in parallel levels l₁, l₂, l₃, etc. Then the relative order of the parallel levels in the space of M is the same as the relative order of the corresponding moments in the time-system α.
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