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\left(\frac{\partial\mathrm{L}}{\partial g^{ab}}\right)_{x}=\frac{1}{2}\sqrt{-g}T_{ab}

in the differentiation on the left hand side the coordinates of the material points are kept constant. To show that T_{ab} and \mathfrak{T}_{c}^{b} satisfy equation (66) we must now show that

-\mathrm{L}-\sqrt{-g}V_{c}^{c}=2\sum(a)g^{ac}\left(\frac{\partial L}{\partial g^{ac}}\right)_{x}

and for b\ne c

-\sqrt{-g}V_{c}^{b}=2\sum(a)g^{ab}\left(\frac{\partial\mathrm{L}}{\partial g^{ac}}\right)_{x}

If here the value (72) is substituted for \mathrm{L} and if (70) is taken into account, these equations say that for all values of b and c we must have

2\sum(a)g^{ab}\left(\frac{\partial H}{\partial g^{ac}}\right)_{x}+V_{c}^{b}=0 (76)

Now this relation immediately follows from a condition, to which \mathrm{L} must be subjected at any rate, viz. that \mathrm{L}dS is a scalar quantity. This involves that in a definite case we must find for H always the same value whatever be the choice of coordinates.

§ 45. Let us suppose that instead of only one coordinate x_{c} a new one x'_{c} has been introduced, which differs infinitely little from x_{c}, with the restriction that if


the term \xi_{c} depends on the coordinate x_b only and is zero at the point in question of the field-figure. The quantities g^{ab} then take other values and in the new system of coordinates the world-lines of the material points will have a slightly changed course.

By each of these circumstances separately H would change, but all together must leave it unaltered. As to the first change we remark that, according to the transformation formula for g^{ab}, the variation \delta g^{ab} vanishes when the two indices are different from c, while

\delta g^{cc}=2g^{cb}\frac{\partial\xi_{c}}{\partial x_{b}}

and for a\ne c

\delta g^{ac}=2g^{ca}=g^{ab}\frac{\partial\xi_{c}}{\partial x_{b}}

The change of H due to these variations is

2\frac{\partial\xi_{c}}{\partial x_{b}}\sum(a)g^{ab}\left(\frac{\partial H}{\partial g^{ac}}\right)_{x}