bears to the unit of velocity [acceleration] on the ratio or according as the motion is parallel to I or perpendicular to I (MVLR).
Let us use F to denote force. Then from the dimensional equation
we shall be able to draw an interesting conclusion concerning the measurement of force.
Suppose that an observer B on a system carries out some observations concerning a certain rectilinear motion, measuring the quantities M', L', T', so that he has the equation
Another observer A on a system (having with respect to the velocity v in the line l) measures the same force, calling it F. Required the value of F in terms of , when the motion is parallel to l and when it is perpendicular to l, the estimate being made by A.
When the motion is perpendicular to l — that is, when the force acts in a line perpendicular to l — we have
When the motion is parallel to l we have
These results may be stated in the following theorem:
Theorem IV. In the same systems of reference as in theorem III., let an observer on measure a given force F' in a direction perpendicular to l and in a direction parallel to l, and let and be the values of this force as measured in the first and second cases respectively by an observer on . Then we have
It is obvious that a similar use may be made of the dimensional equation of any derived unit in determining the relation which exists between this unit in two relatively moving systems of reference.