Page:LorentzGravitation1916.djvu/34

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order. From this we can deduce that is also such a tensor.

Writing for it we find according to (46) and (47) that

is a scalar for every choice of .

This involves that is a covariant tensor of the second order and as the same is true for we must prove the equation

only for one special choice of coordinates.


§ 37. Now this choice can be made in such a way that at the point of the field-figure , , for and that moreover all first derivatives vanish. If then the values at a point near are developed in series of ascending powers of the differences of coordinates the terms directly following the constant ones will be of the second order. It is with these terms that we are concerned in the calculation both of and of for the point . As in the results the coefficients of these terms occur to the first power only, it is sufficient to show that each of the above mentioned terms separately contributes the same value to and to .

From these considerations we may conclude that

(48)

Expressions containing instead of either the variations or might be derived from this by using the relations between the different variations. Of these we shall only mention the formula

(49)


§ 38. In connexion with what precedes we here insert a consideration the purpose of which will be evident later on. Let the infinitely small quantity be an arbitrarily chosen continuous function of the coordinates and let the variations be defined by the condition that at some point the quantities have after the change the values which existed before the change at the point , to which is shifted when is diminished by , while the three other coordinates are left constant. Then we have

and similar formulae for the variations .