Page:LewisTolmanMechanics.djvu/5

Rh and mnm' respectively, and that B's clock runs more slowly than A's. In general, whatever point may be arbitrarily chosen as a point of rest, it will be concluded that any clock in motion relative to this point runs too slowly.

Consider Figure 1 again, assuming system a at rest. We have shown that it is necessary to assume that B's clock runs more slowly than A's in the ratio of the lengths of the path opo to the path mnm' ; in other words, the second of B's clock is longer than the second of A's in the ratio mnm'  to opo. This ratio between the two paths will evidently depend on the relative velocity of the two systems v, and on the velocity of light c.

Obviously from the figure,

$$(op)^{2}=(ln)^{2}=(mn)^{2}-(ml)^{2}$$.

Dividing by (mn)²,

$$\frac{(op)^{2}}{(mn)^{2}}=1-\frac{(ml)^{2}}{(mn)^{2}}$$.

But the distance ml is to the distance mn as v is to c.

Hence

$$\frac{mn}{op}=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$$.

Denoting the important ratio by the letter &beta;, we see that in general a second measured by a moving clock bears to a second measured by a stationary clock the ratio $$\frac{1}{\sqrt{1-\beta^{2}}}$$.

Whatever assumption the observers A and B may make as to their motion, it is obvious that their measurements of length, at least in a direction perpendicular to their line of relative motion, will lead to no disagreement. For evidently, if each observer with a measuring rod determines the distance from his system to the other, the two determinations must agree. Otherwise' the condition of symmetry required by the principle of relativity would not be fulfilled.

But let us now consider distances parallel to the line of relative motion.

A system (Figure 2) has a source of light at m and a reflecting mirror at n. If we consider the whole system to be at absolute rest, it is evident that a light signal sent from m to the mirror, and reflected back,