The dx′σ’s are expressed as linear and homogeneous function of dxν’s; we can look upon the differentials of the co-ordinates as the components of a tensor, which we designate specially as a contravariant Four-vector. Everything which is defined by Four quantities Aσ, with reference to a co-ordinate system, and transforms according to the same law,
"(5a)."
we may call a contra-variant Four-vector. From (5. a), it follows at once that the sums (Aσ ± Bσ) are also components of a four-vector, when Aσ and Bσ 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 Aν as the components of a covariant four-vector, when for any choice of the contra-variant four vector Bν (6) ∑ν Aν Bν = 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 A′σ Bσ′ = ∑ν Aν Bν.
the expressions
for Bν following from the inversion of the equation (5a) we get