Page:PoincareDynamiqueJuillet.djvu/41

it better, than the 2 expressions:

$x^{2}+y^{2}+z^{2}-t^{2},\ x\delta x+y\delta y+z\delta z-t\delta t$

or the 4 expressions of the same form, deduced from permuting (in an arbitrary way) the three points P, P', P".

But what we look for are the functions of 10 variables (2) that are invariants; so we must, among the combinations of our 6 invariants, seek those which depend only on these 10 variables, that is to say those that are homogeneous of degree 0 as compared to δx, δy, δz, δt, as compared to δ₁x, δ₁y, δ₁z, δ₁t. We will thus have 4 distinct invariants, which are:

(5)	$\sum x^{2}-t^{2},\quad {\frac {t-\sum x\xi }{\sqrt {1-\sum \xi ^{2}}}},\quad {\frac {t-\sum x\xi _{1}}{\sqrt {1-\sum \xi _{1}^{2}}}},\quad {\frac {t-\sum \xi \xi _{1}}{\sqrt {\left(1-\sum \xi ^{2}\right)\left(1-\sum \xi _{1}^{2}\right)}}}$

Let us now consider the transformations undergone by the components of the force; resume the equations (11) of § 1, which relate not to the force X₁, Y₁, Z₁, which we consider here, but to the force X, Y, Z referred to unit volume. We pose also:

$T=\sum X\xi ;$

we see that these equations (11) can be written as (l = 1):

(6)	${\begin{cases}X^{\prime }=k(X+\epsilon T),\quad &T^{\prime }=k(T+\epsilon X),\\\\Y^{\prime }=Y,&Z^{\prime }=Z;\end{cases}}$

so that X, Y, Z, T undergo the same transformation as x, y, z, t. The invariants of the group are therefore

$\sum X^{2}-T^{2},\quad \sum Xx-Tt,\quad \sum X\delta x-T\delta t,\quad \sum X\delta _{1}x-T\delta _{1}t.$

But this is not X, Y, Z which we need, it is X₁, Y₁, Z₁ with

$T_{1}=\sum X_{1}\xi .$

We see that

${\frac {X_{1}}{X}}={\frac {Y_{1}}{Y}}={\frac {Z_{1}}{Z}}={\frac {T_{1}}{T}}={\frac {1}{\rho }}.$

So the Lorentz transformations act on X₁, Y₁, Z₁, T₁ in the same manner as X, Y, Z, T, with the difference that these expressions are also multiplied by

${\frac {\rho }{\rho ^{\prime }}}={\frac {1}{k(1+\xi \epsilon )}}={\frac {\delta t}{\delta t^{\prime }}}.$

Similarly it would act on ξ, η, ζ, 1, in the same manner as δx, δy, δz, δt, with the difference that these expressions are also multiplied by the same factor:

${\frac {\delta t}{\delta t^{\prime }}}={\frac {1}{k(1+\xi \epsilon )}}.$

Consider then $X,\ Y,\ Z,\ T{\sqrt {-1}}$ as the coordinates of a fourth point