Page:Grundgleichungen (Minkowski).djvu/52

 In consequence of this condition, the integral (7) taken over the whole range of the sickle, varies on account of the displacement as a definite function $$\mathsf{N}+\delta\mathsf{N}$$ of $$\vartheta$$, and we may call this function $$\mathsf{N}+\delta\mathsf{N}$$ as the mass action of the virtual displacement.

If we now introduce the method of writing with indices, we shall have

Now on the basis of the remarks already made, it is clear that the value of $$\mathsf{N}+\delta\mathsf{N}$$, when the value of the parameter is $$\vartheta$$, will be: —

the integration extending over the whole sickle $$d(\tau + \delta\tau)$$, where $$d(\tau + \delta\tau)$$ denotes the magnitude, which is deduced from

$$\sqrt{-(dx_{1}+d\delta x_{1})^{2}-(dx_{2}+d\delta x_{2})^{2}-(dx_{3}+d\delta x_{3})^{2}-(dx_{4}+d\delta x_{4})^{2}}$$

by means of (9) and

$$dx_{1}=w_{1}d\tau,\ dx_{2}=w_{2}d\tau,\ dx_{3}=w_{3}d\tau,\ dx_{4}=w_{4}d\tau,\ d\vartheta=0$$

therefore: —

We shall now subject the value of the differential quotient

to a transformation. Since each $$\delta x_{h}$$, as a function of $$x_{1},\ x_{2},\ x_{3},\ x_{4},\ \vartheta$$ vanishes for the zero-value of the paramater $$\vartheta$$, so in general $$\frac{\partial\delta x_{h}}{\partial x_{k}}=0$$ for $$\vartheta=0$$.

Let us now put

then on the basis of (10) and (11), we have the expression (12):

$$-\int\int\int\int\ \nu\underset{h}{\sum}w_{h}\left(\frac{\partial\xi_{h}}{\partial x_{1}}w_{1}+\frac{\partial\xi_{h}}{\partial x_{2}}w_{2}+\frac{\partial\xi_{h}}{\partial x_{3}}w_{3}+\frac{\partial\xi_{h}}{\partial x_{4}}w_{4}\right)dx\ dy\ dz\ dt$$.

for the system $$x_{1},\ x_{2},\ x_{3},\ x_{4}$$ on the boundary of the sickle, $$\delta x_{1},\ \delta x_{2},\ \delta x_{3},\ \delta x_{4}$$ shall vanish for every value of $$\vartheta$$