Page:PoincareDynamiqueJuillet.djvu/33

(7)	$h^{\prime -1}{\frac {d\eta ^{\prime }}{dt^{\prime }}}+h^{\prime -3}\eta ^{\prime }M^{\prime }=\left[h^{-1}{\frac {d\eta }{dt}}+h^{-3}\eta M\right]\mu ^{-1}h^{-1}.$

Let us return now to equations (11^bis) of § 1; we can regard X₁, Y₁, Z₁ as having the same meaning as in equations (5). On the other hand, we have l = 1 and ${\frac {\rho ^{\prime }}{\rho }}=k\mu$; these equations then become:

(8)	${\begin{cases}X_{1}^{\prime }=\mu ^{-1}\left(X_{1}+\epsilon \sum X_{1}\xi \right),\\\\Y_{1}^{\prime }=k^{-1}\mu ^{-1}Y_{1}.\end{cases}}$

We calculate ΣX₁ξ using equation (5), we find:

$\Sigma X_{1}\xi =h^{-3}M,$

where:

(9)	${\begin{cases}X_{1}^{\prime }=\mu ^{-1}\left(X_{1}+\epsilon h^{-3}M\right),\\\\Y_{1}^{\prime }=k^{-1}\mu ^{-1}Y_{1}.\end{cases}}$

Comparing equations (5) (6), (7) and (9), we finally find:

(10)	${\begin{cases}h^{\prime -1}{\frac {d\xi ^{\prime }}{dt^{\prime }}}+h^{\prime -3}\xi ^{\prime }M^{\prime }=X_{1}^{\prime },\\\\h^{\prime -1}{\frac {d\eta ^{\prime }}{dt^{\prime }}}+h^{\prime -3}\eta ^{\prime }M^{\prime }=Y_{1}^{\prime }\end{cases}}$

This shows that the equations of quasi-stationary motion are not altered by the Lorentz transformation, but it still does not prove that the hypothesis of Lorentz is the only one that leads to this result.

To establish this point, we will restrict ourselves, as Lorentz did, to certain particular cases; it will be obviously sufficient for us to show a negative proposal.

How do we first extend the hypotheses underlying the above calculation?

1° Instead of assuming l = 1 in the Lorentz transformation, we assume any l.

2° Instead of assuming that F is proportional to the volume, and hence that H is proportional to h, we assume that F is any function of θ and r, so that [after replacing θ and r with their values as functions of V, from the first two equations (1)] H is any function of V.

I note first that, assuming H = h, we must have l = 1; and in fact the equations (6) and (7) remain, except that the right-hand sides will be multiplied by ${\tfrac {1}{l}}$; so do equations (9), except that the right-hand sides will be multiplied by ${\tfrac {1}{l^{2}}}$; and finally the equations (10), except that the right-hand sides will be multiplied by ${\tfrac {1}{l}}$. If we want that the equations of motion are not altered by the Lorentz transformation