Page:CarmichealMass.djvu/9

$S_{1}$ bears to the unit of velocity [acceleration] on $S_{2}$ the ratio $1:1\left[1:{\sqrt {1-\beta ^{2}}}\right]$ or $1:{\sqrt {1-\beta ^{2}}}\left[1:1-\beta ^{2}\right]$ 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

$F={\frac {ML}{T^{2}}}$,

we shall be able to draw an interesting conclusion concerning the measurement of force.

Suppose that an observer B on a system $S_{2}$ carries out some observations concerning a certain rectilinear motion, measuring the quantities M', L', T', so that he has the equation

$F'={\frac {M'L'}{T'^{2}}}$.

Another observer A on a system $S_{1}$ (having with respect to $S_{2}$ the velocity v in the line l) measures the same force, calling it F. Required the value of F in terms of $F'$, 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

$F_{1}={\frac {ML}{T^{2}}}={\frac {M'{\sqrt {1-\beta ^{2}}}\cdot L'}{T'^{2}\left(1-\beta ^{2}\right)}}={\frac {F'}{\sqrt {1-\beta ^{2}}}}$

When the motion is parallel to l we have

$F_{2}={\frac {ML}{T^{2}}}={\frac {M'\left(1-\beta ^{2}\right)^{\frac {3}{2}}\cdot L{\sqrt {1-\beta ^{2}}}}{T'^{2}\left(1-\beta ^{2}\right)}}=\left(1-\beta ^{2}\right)F'.$

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 $S_{2}$ measure a given force F' in a direction perpendicular to l and in a direction parallel to l, and let $F_{1}$ and $F_{2}$ be the values of this force as measured in the first and second cases respectively by an observer on $S_{1}$. Then we have

$F_{1}={\frac {F'}{\sqrt {1-\beta ^{2}}}},\ F_{2}=\left(1-\beta ^{2}\right)F'$

$\scriptstyle \left(MVLRC_{1}C_{2}\right).$

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.