in the differentiation on the left hand side the coordinates of the material points are kept constant. To show that and satisfy equation (66) we must now show that
If here the value (72) is substituted for and if (70) is taken into account, these equations say that for all values of and we must have
Now this relation immediately follows from a condition, to which must be subjected at any rate, viz. that is a scalar quantity. This involves that in a definite case we must find for always the same value whatever be the choice of coordinates.
§ 45. Let us suppose that instead of only one coordinate a new one has been introduced, which differs infinitely little from , with the restriction that if
the term depends on the coordinate only and is zero at the point in question of the field-figure. The quantities 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 would change, but all together must leave it unaltered. As to the first change we remark that, according to the transformation formula for , the variation vanishes when the two indices are different from , while
The change of due to these variations is