Page:PoincareDynamiqueJuillet.djvu/15

 so that if we set

$$J^{\prime}=\int dt^{\prime}\ d\tau^{\prime}\left(\frac{\sum f^{\prime2}}{2}-\frac{\sum\alpha^{\prime2}}{2}\right),$$

we get:

J' = J.

However, to justify this equality, the integration limits have to be the same; so far we have assumed that t varies from t0 to t1, and x, y, z from ∞ to + ∞. On this account the integration limits would be affected by the transformation, but nothing prevents us from assuming t0 =- ∞, t1 = + ∞; with those conditions the limits are the same for J and J'.

We then compare the following two equations analogues to equation (10) of § 2:

{{MathForm1|(2)|$$\left\{ \begin{align} \delta J & =-\int\sum X\delta U\ d\tau\ dt\\ \delta J' & =-\int\sum X^{\prime}\delta U^{\prime}\ d\tau^{\prime}\ dt^{\prime}. \end{align}\right.$$}}

For this, we must first compare δU with δU.

Consider an electron whose initial coordinates are x0, y0, z0; its coordinates at the instant t are

$$x=x_{0}+U,\quad y=y_{0}+V,\quad z=z_{0}+W.$$

If one considers the electron after the corresponding transformation, it will have as coordinates

$$x^{\prime}=kl\left(x+\epsilon t\right),\quad y^{\prime}=ly,\quad z^{\prime}=lz$$

where

$$x^{\prime}=x_{0}+U^{\prime},\quad y^{\prime}=y_{0}+V^{\prime},\quad z^{\prime}=z_{0}+W^{\prime}$$

but it will only attain these coordinates at the instant

$$t^{\prime}=kl\left(t+\epsilon x\right).$$

If we subject our variables to the variations δU, δV, δW, and when we give at the same time t an increasement &delta;t, the coordinates x, y, z will experience a total increasement

$$\delta x=\delta U+\xi\delta t,\quad\delta y=\delta V+\eta\delta t,\quad\delta z=\delta W+\zeta\delta t.$$

We will also have:

$$\delta x^{\prime}=\delta U^{\prime}+\xi^{\prime}\delta t^{\prime},\quad\delta y^{\prime}=\delta V^{\prime}+\eta^{\prime}\delta t^{\prime},\quad\delta z^{\prime}=\delta W^{\prime}+\zeta^{\prime}\delta t^{\prime},$$

and in virtue of the transformation:

$$\delta x^{\prime}=kl\left(\delta x+\epsilon\delta t\right),\quad\delta y^{\prime}=l\delta y,\quad\delta z^{\prime}=l\delta z,\quad\delta t^{\prime}=kl\left(\delta t+\epsilon\delta x\right).$$

hence, assuming δt = 0, the relations:

{{MathForm1||$$\left\{ \begin{align} \delta x^{\prime} & =\delta U^{\prime}+\xi^{\prime}\delta t^{\prime} & =kl\delta U,\\ \delta y^{\prime} & =\delta V^{\prime}+\eta^{\prime}\delta t^{\prime} & =l\delta V,\\ \delta t^{\prime} & =kl\epsilon\delta U. \end{align}\right.$$}}