we may call a contra-variant Four-vector. From (5. a), it follows at once that the sums (A^{σ} ± B^{σ}) are also components of a four-vector, when A^{σ} and B^{σ} are so; corresponding relations hold also for all systems afterwards introduced as "tensors" (Rule of addition and subtraction of Tensors).
Co-variant Four-vector.
We call four quantities A_{ν} as the components of a covariant four-vector, when for any choice of the contra-variant four vector B^{ν} (6) Σ_{ν} A_{ν} B^{ν} = Invariant. From this definition follows the law of transformation of the co-variant four-vectors. If we substitute in the right hand side of the equation
Σ_{σ} A´_{σ} B^{σ´}}? more below] = Σ_{ν} A_{ν} B^ν.
the expressions
Σ_{σ} [part]x_{ν}/[part]x_{σ´} B^{σ´}
for B^{ν} following from the inversion of the equation (5a) we get
Σ_{σ} B^{σ´} Σ_{ν} [part]x_{ν}/[part]x_{σ´} A_{ν} = Σ_{σ} B^{σ´} A´_{σ}
As in the above equation B^{σ´} are independent of one another and perfectly arbitrary, it follows that the transformation law is:—
A´_{σ} = Σ [part]x_{ν}/[part]x_{σ´} A_{ν}.
