the tensor. There are now certain rules according to which the components can be calculated in a new system of co-ordinates, when these are known for the original system, and when the transformation connecting the two systems is known. The things herefrom designated as "Tensors" have further the property that the transformation equation of their components are linear and homogeneous; so that all the components in the new system vanish if they are all zero in the original system. Thus a law of Nature can be formulated by putting all the components of a tensor equal to zero so that it is a general co-variant equation; thus while we seek the laws of formation of the tensors, we also reach the means of establishing general co-variant laws.
5. Contra-variant and co-variant Four-vector.
Contra-variant Four-vector. The line-element is defined by the four components dx_{ν}, whose transformation law is expressed by the equation
(5) dx´_{σ} = Σ_{ν} [part]x´_{σ}/[part]x_{ν} dx_{nu}.
The dx´_{σ}'s are expressed as linear and homogeneous function of dx_{ν}'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 A^σ, with reference to a co-ordinate system, and transforms according to the same law,
(5a) A^σ = Σ_{nu} [part]x´_{σ}/[part]x_{ν} A^ν
