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:

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''

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.