Page:LorentzGravitation1915.djvu/9

 To these equations we add the transformation formula for $$\bar{\psi'}_{ab}$$, which may be derived from (28)

§ 9. We shall now consider the 6 quantities (27) which we shall especially call "the quantities $$\psi$$" and the corresponding quantities $$\bar{\psi}$$, viz. $$\bar{\psi}_{41}\dots\bar{\psi}_{12}$$.

According to (30) these latter are homogeneous and linear functions of the former and as (because of (5)) the coefficient of $$\psi_{cd}$$ in $$\bar{\psi}_{ab}$$ is equal to the coefficient of $$\psi_{ab}$$ in $$\bar{\psi}_{cd}$$, there exists a homogeneous quadratic function $$\mathrm{L}$$ of $$\psi_{41}\dots\psi_{12}$$, which, when differentiated with respect to these quantities, gives $$\psi_{41}\dots\psi_{12}$$. Therefore

and

If, as in (34), we have to consider derivatives of $$\mathrm{L}$$, this quantity will be regarded as a quadratic function of the quantities $$\psi$$.

The quantity $$\mathrm{L}$$ can now play the same part as the quantity that is represented by the same letter in §§ 4 — 6. Again $$\mathrm{L}dS$$ is invariant when the coordinates are changed.