Page:Elektrische und Optische Erscheinungen (Lorentz) 071.jpg

cannot occur in the relations between ${\overline {\mathfrak {E}}}$ and ${\mathfrak {M}}$. To indicate this, we replace (58) for resting bodies by

${\overline {\mathfrak {E}}}=F_{1}({\mathfrak {M,{\ddot {M}},}}\ldots )$

(59)

If we again allow the translation, then we have to add to $F_{1}$ still another vector, whose components are linear and homogeneous functions of ${\mathfrak {M,{\dot {M}},{\ddot {M}}}}$, ..., and which contain in any term one of the factors ${\mathfrak {p}}_{x},{\mathfrak {p}}_{y},{\mathfrak {p}}_{z}$; also this new vector must stay unchanged when passing to the inverse motion. As in this case the components ${\mathfrak {p}}_{x},{\mathfrak {p}}_{y},{\mathfrak {p}}_{z}$ contain opposite signs, thus they can only be multiplied by such magnitudes which also change the sign, i.e. by uneven derivatives with respect to time. The equations (58) therefore generally assume the form

${\overline {\mathfrak {E}}}=F_{1}({\mathfrak {M,{\ddot {M}},}}\ldots )+F_{2}({\mathfrak {M,{\dot {\ddot {M}}},}}\ldots {\mathfrak {p}})$

(60)

An additional simplifications we can achieved, by considering a certain kind of homogeneous light, i.e. by considering goniometric functions of time of a certain period T. Then

${\ddot {\mathfrak {M}}}=-\left({\frac {2\pi }{T}}\right)^{2}{\mathfrak {M}},\ {\dot {\ddot {\mathfrak {M}}}}=-\left({\frac {2\pi }{T}}\right)^{2}{\dot {\mathfrak {M}}}$, etc.

(61)

If we in (60) express in this way all even derivatives by ${\mathfrak {M}}$ and all uneven by ${\dot {\mathfrak {M}}}$, it will be given

${\overline {\mathfrak {E}}}=F_{1}({\mathfrak {M}})+F_{2}({\mathfrak {{\dot {M}},p}})$

(62)

The components of $F_{1}$ are now homogeneous functions of ${\mathfrak {M}}_{x},{\mathfrak {M}}_{y},{\mathfrak {M}}_{z}$ and its derivatives with respect to x, y, z, while $F_{2}$ depends in a similar way on ${\dot {\mathfrak {M}}}$. The coefficients of this function may well depend on the oscillation period T, since we have introduced the values (61) into (60).