Page:PoincareDynamiqueJuillet.djvu/47

so that the total force can be divided into three components corresponding to the three brackets of expression (12); the first component has a vague analogy with the mechanical force due to the electric field, the other two with mechanical forces due to a magnetic field; to complete the analogy I can, under the first point, replace ${\tfrac {1}{B^{3}}}$ by ${\tfrac {C}{B^{3}}}$ in equations (11), so that X₁, Y₁, Z₁ only depend linearly on the velocity ξ, η, ζ of the attracted body, since C has disappeared from the denominator of (11^bis).

We pose then:

(13)

${\begin{cases}k_{1}(x+r\xi _{1})=\lambda ,\quad k_{1}(y+r\eta _{1})=\mu ,\quad k_{1}(z+r\zeta _{1})=\nu ,\\\\k_{1}\left(\eta _{1}z-\zeta _{1}y\right)=\lambda ^{\prime },\quad k_{1}\left(\zeta _{1}x-\xi _{1}z\right)=\mu ^{\prime },\quad k_{1}\left(\xi _{1}y-x\eta _{1}\right)=\nu ^{\prime };\end{cases}}$

it follows that C had disappeared from the denominator of (11a):

(14)

${\begin{cases}X_{1}={\frac {\lambda }{B^{3}}}+{\frac {\eta \nu ^{\prime }-\zeta \mu ^{\prime }}{B^{3}}},\\\\Y_{1}={\frac {\mu }{B^{3}}}+{\frac {\zeta \lambda ^{\prime }-\xi \nu ^{\prime }}{B^{3}}},\\\\Z_{1}={\frac {\nu }{B^{3}}}+{\frac {\xi \mu ^{\prime }-\eta \lambda ^{\prime }}{B^{3}}},\end{cases}}$

and there will also:

(15)	$B^{2}=\sum \lambda ^{2}-\sum \lambda ^{\prime 2}.$

Then λ, μ, ν or ${\tfrac {\lambda }{B^{3}}},\ {\tfrac {\mu }{B^{3}}},\ {\tfrac {\nu }{B^{3}}}$ is a kind of electric field, while λ', μ', ν' or rather ${\tfrac {\lambda ^{\prime }}{B^{3}}},\ {\tfrac {\mu ^{\prime }}{B^{3}}},\ {\tfrac {\nu ^{\prime }}{B^{3}}}$ is a kind of magnetic field.

3° The postulate of relativity would require us to adopt solution (11) or solution (14) or any solution that would inferred by using the first remark; but the first question that arises is whether they are compatible with astronomical observations; the discrepancy with Newton's law is of the order ξ², that is to say, 10000 times smaller when it were of order ξ, that is to say, if the propagation happens with the speed of light, ceteris non mutatis; it is permissible to hope that it will not be too great. But only a thorough discussion will be able to teach it to us.

Paris, July 1905.

H. Poincaré