Page:The principle of relativity (1920).djvu/178

 ''Remarks on the simplification of the mode of writing the expressions.'' A glance at the equations of this paragraph will show that the indices which appear twice within the sign of summation [for example ν in (5)] are those over which the summation is to be made and that only over the indices which appear twice. It is therefore possible, without loss of clearness, to leave off the summation sign; so that we introduce the rule: wherever the index in any term of an expression appears twice, it is to be summed over all of them except when it is not expressedly said to the contrary.

The difference between the co-variant and the contra-*variant four-vector lies in the transformation laws [(7) and (5)]. Both the quantities are tensors according to the above general remarks; in it lies its significance. In accordance with Ricci and Levi-civita, the contravariants and co-variants are designated by the over and under indices.

§ 6. Tensors of the second and higher ranks.

Contravariant tensor:—If we now calculate all the 16 products A^{μν} of the components A^{μ} B^{ν}, of two contravariant four-vectors

(8) A^{μν} = A^{μ}B^{ν}

A^{μν}, will according to (8) and (5 a) satisfy the following transformation law.

(9) A^{στ´} = [part]x´_{sigma}/[part]x_{mu} [part]x´_{tau}/[part]x_{nu} A^{μν}

We call a thing which, with reference to any reference system is defined by 16 quantities and fulfils the transformation relation (9), a contravariant tensor of the second