Page:Zur Dynamik bewegter Systeme.djvu/16

 Now, by (13) and (14) we have:

and:

$$\frac{dt'}{dt}=\frac{c^{2}-v\dot{x}}{c\sqrt{c^{2}-v^{2}}}$$.

It follows:

$$d\frac{\partial}{\partial\dot{y}}\left(H'\sqrt{\frac{c^{2}-q^{2}}{c^{2}-q'^{2}}}\right)=d\frac{\partial H}{\partial\dot{y}}$$

and by integration:

The constant of integration, an absolute constant, vanishes because only q' = q H' goes over into H.

§ 9. Now, the four equations (19) and (24) give by integration:

$$H'\sqrt{\frac{c^{2}-q^{2}}{c^{2}-q'^{2}}}=H+const$$.

The constant does not depend on V, T, $$\dot{y},\dot{z}$$; but it can still depend on $$\dot{x}$$, or by (14), $$\frac{c^{2}-q^{2}}{c^{2}-q'^{2}}$$. We therefore write:

$$H'\sqrt{\frac{c^{2}-q^{2}}{c^{2}-q'^{2}}}=H+f\left(\frac{c^{2}-q^{2}}{c^{2}-q'^{2}}\right)$$

and determine the most general expression of the function f.

At first, we have:

Since the function H only depends on q, V and T, and since V' and T' are only connected to V and T by the relations (17), then the right-hand side of the equation as well as the left-hand side, are of the form:

$$\frac{1}{\sqrt{c^{2}-q^{2}}}\cdot f\left(\frac{c^{2}-q^{2}}{c^{2}-q'^{2}}\right)=Q'-Q$$,