Page:PoincareDynamiqueJuillet.djvu/17

and in addition

$x^{\prime \prime }=k^{\prime }l^{\prime }(x^{\prime }+\epsilon ^{\prime }t^{\prime }),\quad y^{\prime \prime }=l^{\prime }y^{\prime },\quad z^{\prime \prime }=l^{\prime }z^{\prime },\quad t^{\prime \prime }=k^{\prime }l^{\prime }(t^{\prime }+\epsilon ^{\prime }x^{\prime }),$

with

$k^{-2}=1-\epsilon ^{2},\quad k^{\prime -2}=1-\epsilon ^{\prime 2}$

it follows:

$x^{\prime \prime }=k^{\prime \prime }l^{\prime \prime }(x+\epsilon ^{\prime \prime }t),\quad y^{\prime \prime }=l^{\prime \prime }y,\quad z^{\prime \prime }=l^{\prime \prime }z,\quad t^{\prime \prime }=k^{\prime \prime }l^{\prime \prime }(t+\epsilon ^{\prime \prime }x),$

with

$\epsilon ^{\prime \prime }={\frac {\epsilon +\epsilon ^{\prime }}{1+\epsilon \epsilon ^{\prime }}},\quad l^{\prime \prime }=ll^{\prime },\quad k^{\prime \prime }=kk^{\prime }(1+\epsilon \epsilon ^{\prime })={\frac {1}{\sqrt {1-\epsilon ^{\prime \prime 2}}}}.$

If we take for l the value 1, and we suppose ε infinitely small,

$x^{\prime }=x+\delta x,\quad y^{\prime }=y+\delta y,\quad z^{\prime }=z+\delta z,\quad t^{\prime }=t+\delta t$

it follows:

$\delta x=\epsilon t,\quad \delta y=\delta z=0,\quad \delta t=\epsilon x.$

This is the infinitesimal generator of the transformation group, which I call the transformation T₁, and which can be written in Lie's notation:

$t{\frac {d\varphi }{dx}}+x{\frac {d\varphi }{dt}}=T_{1}.$

If we assume ε = 0 and l = 1 + δl, we find instead

$\delta x=x\delta l,\quad \delta y=y\delta l,\quad \delta z=z\delta l,\quad \delta t=t\delta l$

and we would have another infinitesimal transformation t₀ of the group (assuming that l and ε are regarded as independent variables) and we would have with Lie's notation:

$T_{0}=x{\frac {d\varphi }{dx}}+y{\frac {d\varphi }{dy}}+z{\frac {d\varphi }{dz}}+t{\frac {d\varphi }{dt}}.$

But we could give the y- or z-axes the special role, which we gave the x-axis; thus we have two further infinitesimal transformations:

$T_{2}=t{\frac {d\varphi }{dy}}+y{\frac {d\varphi }{dt}},$ $T_{3}=t{\frac {d\varphi }{dz}}+z{\frac {d\varphi }{dt}},$

which also would not alter the equations of Lorentz.

We can form combinations devised by Lie, such as

$\left[T_{1},T_{2}\right]=x{\frac {d\varphi }{dy}}-y{\frac {d\varphi }{dx}},$

but it is easy to see that this transformation is equivalent to a coordinate change, the axes are rotating a very small angle around the z-axis. We