and equating the result to zero, we obtain an equation which corresponds to relations holding in three-dimensional space. This equation is given by
where the
’s are the components of this velocity along the three mutually perpendicular directions of space. When a mathematical expression of this kind is given as invariant or absolute, the theory of groups enables us to determine the nature of the geometrical rules according to which the variables entering into the expression may be added together. In the present case these variables represent velocities, so that we are on the way to discover the rules governing the addition of velocities. From this equation the following results are obtained.
1°. If
, the invariant velocity, is infinite, as in classical science, it is found that the composition of velocities must be Euclidean; so we should be able to combine velocities graphically by the well-known rule of the parallelogram or of the Euclidean triangle. In this case, of course, velocities which lie in the same direction can be added up and subtracted like numbers.
2°. If