By the same method we can show that $A_{\mu}B^{\mu}$, $A_{\mu\nu}^{\mu\nu}$, $A_{\mu}^{\nu}B_{\nu}^{\mu}$ are invariants. In general when an upper and lower suffix are the same the corresponding covariant and contravariant qualities cancel out. If all suffixes cancel out in this way, the expression must be invariant. The identified suffixes must be of opposite characters; the expression $A_{\mu\sigma\sigma}^{\tau}$ is not a tensor, and no interest is attached to it. We see that the suffixes keep a tally of what we have called the generalised dimensions of the terms in our equations. After cancelling out any suffixes which appear in both upper and lower positions, the remaining suffixes must appear in the same position in each term of an equation. When that is satisfied each term will undergo the same set of operations when a transformation of coordinates is made, and the equation will continue to hold in all \PageSep{54} systems of coordinates. This may be compared with the well-known condition that each term must have the same physical dimensions, so that it undergoes multiplication by the same factor when a change of units is made and the equation continues to hold in all systems of units. Just as we can infer the physical dimensions of some novel entity entering \index{Dimensions, principle of}% \index{Principle!of dimensions}% into a physical equation, so we can infer the contravariant and covariant dimensions of an expression whose character was hitherto unknown. For example, if the equation is \[ A(\mu\nu) B_{\nu\sigma} = C_{\mu\sigma}, \Tag{(24.1)} \] where the nature of $A(\mu\nu)$ is not known initially, we see that $A(\mu\nu)$~must be a tensor of the character~$A_{\mu}^{\nu}$, so as to give \[ A_{\mu}^{\nu} B_{\nu\sigma} = C_{\mu\sigma}, \] which makes the covariant dimensions on both sides consistent. The equation~\Eq{(24.1)} may be written symbolically \[ A(\mu\nu) = C_{\mu\sigma}/B_{\nu\sigma}, \] and the conclusion is that not only the product but also the (symbolic) quotient of two tensors is a tensor. Of course, the operation here indicated is not that of ordinary division. This quotient law is a useful aid in detecting the tensor-character of \index{Quotient law}% expressions. It is not claimed that the general argument here given amounts to a strict mathematical proof. In most cases we can supply the proof required by one or more applications of the following rigorous theorem--- A quantity which on inner multiplication by \emph{any} covariant (alternatively, by \emph{any} contravariant) vector always gives a tensor, is itself a tensor.