The fundamental idea of the general co-variant theory is this:—With reference to any co-ordinate system, let certain things (tensors) be defined by a number of functions of co-ordinates which are called the components of 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) $$ dx'_{\sigma} = \sum_{\nu} \frac{\partial x'_{\sigma}}{\partial x_{\nu}} dx_{\nu} $$ 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) $$ A^{\sigma} = \sum_{\nu} \frac{\partial x'_{\sigma}}{\partial x_{\nu}} A^{\nu} $$ 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) ∑_{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} A′_{σ} B^{σ′} = ∑_{ν} A_{ν} B^{ν}. the expressions $$ \sum_{\sigma} \frac{\partial x_{\nu}}{\partial x_{\sigma'}} B^{\sigma'} $$ for B^{ν} following from the inversion of the equation (5a) we get $$ \sum_{\sigma} B^{\sigma'} \sum_{\nu} \frac{\partial x_{\nu}}{\partial x_{\sigma'}} A_{\nu} = \sum_{\sigma} B^{\sigma'} A'_{\sigma} $$ As in the above equation B^{σ′} are independent of one another and perfectly arbitrary, it follows that the transformation law is:— $$ A'_{\sigma} = \sum \frac{\partial x_{\nu}}{\partial x_{\sigma'}} A_{\nu} $$