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§ 54. The presupposition, that no derivatives with respect to x, y, z occur, has led us to equation (65), from which the rotation of the polarization plane does not arise. Thus it is necessary, as it was already indicated earlier to assume (at least in the expression $F_{1}({\mathfrak {M}})$) derivatives with respect to the coordinates. The most simple is, to add to the second term of (65) another vector ${\mathfrak {R}}$, whose components do linearly and homogeneously depend on the first derivatives of ${\mathfrak {M}}_{x},\ {\mathfrak {M}}_{y},\ {\mathfrak {M}}_{z}$. Magnitude and direction will now again be closely determined by isotropy. Namely, if we imagine at any point of space a line, that represents the vector ${\mathfrak {M}}$, and in addition in the considered point the vector ${\mathfrak {R}}$, then after an arbitrary rotation of that entire figure, ${\mathfrak {R}}$ must still fit to ${\mathfrak {M}}$. Only the assumption^[1]

${\mathfrak {R}}=j\ Rot\ {\mathfrak {M}}$,

is in agreement with this,

1. After a rotation of the mentioned figure we want, as we are really free to do this, to apply again the original coordinate axis for the decomposition of the vectors and the formation of the derivatives. At first, only a rotation of 180° around the axis takes place. Here, ${\mathfrak {R}}_{x}$ remains unchanged; consequently in the expression for this component only these derivatives of ${\mathfrak {M}}_{x},\ {\mathfrak {M}}_{y},\ {\mathfrak {M}}_{z}$ can occur, which do not change the sign. These are
   ${\frac {\partial {\mathfrak {M}}_{x}}{\partial x}},\ {\frac {\partial {\mathfrak {M}}_{y}}{\partial y}},\ {\frac {\partial {\mathfrak {M}}_{y}}{\partial z}},\ {\frac {\partial {\mathfrak {M}}_{z}}{\partial y}},\ {\frac {\partial {\mathfrak {M}}_{z}}{\partial z}}$.

   If we further notice, that in the course of a rotation of 180° around the y- or z-axis, ${\mathfrak {R}}_{x}$ assumes the opposite direction, and that also those derivatives are excluded, which retain the same sign during one of these rotations, the we find, that ${\mathfrak {R}}_{x}$ must be of the form

   $j{\frac {\partial {\mathfrak {M}}_{z}}{\partial y}}+j{\frac {\partial {\mathfrak {M}}_{y}}{\partial z}}$

   Eventually we imagine still another rotation around 90° around the x-axis, whereby OY is transformed into OZ. After that rotation, ${\frac {\partial {\mathfrak {M}}_{z}}{\partial y}}$ and ${\frac {\partial {\mathfrak {M}}_{y}}{\partial z}}$ have the values, that previously belonged to $-{\frac {\partial {\mathfrak {M}}_{y}}{\partial z}}$ and $-{\frac {\partial {\mathfrak {M}}_{z}}{\partial y}}$; however, since ${\mathfrak {R}}_{x}$ hasn't changed, then $j'=-j$. From ${\mathfrak {R}}_{x}$ we find ${\mathfrak {R}}_{y}$ and ${\mathfrak {R}}_{z}$ by permutation of the letters.