obtain the tensor T from the tensors A and B of different kinds:—
T_{μνσ} = A_{μν}B_{σ},
T^{αβγδ} = A^{αβ}B_{γδ},
T_{αβ}^{γδ} = A_{αβ}B^{γδ}.
The proof of the tensor character of T, follows immediately from the expressions (8), (10) or (12), or the transformation equations (9), (11), (13); equations (8), (10) and (12) are themselves examples of the outer multiplication of tensors of the first rank.
Reduction in rank of a mixed Tensor.
From every mixed tensor we can get a tensor which is
two ranks lower, when we put an index of co-variant
character equal to an index of the contravariant character
and sum according to these indices (Reduction). We get
for example, out of the mixed tensor of the fourth rank
A_{αβ}^{γδ}, the mixed tensor of the second rank
A_{β}^{δ} = A_{αβ}^{αδ} = ([Sum]_α A_{αβ}^{αδ})
and from it again by "reduction" the tensor of the zero rank
A = A_{β}^{β} = A_{αβ}^{αβ}.
The proof that the result of reduction retains a truly tensorial character, follows either from the representation
