It may be remarked that if we take a force $(X, Y, Z)$ and transform it to \index{Force, covariant and contravariant components}% polar coordinates, whether as a covariant or a contravariant vector, in neither case do we obtain the quantities called polar components in elementary mechanics. The latter are not in our view the true polar components; they are merely rectangular components in three new directions, viz.\ radial and transverse. In general the elementary definitions of physical quantities do not contemplate other than rectangular components, and they may need to be supplemented before we can decide whether the physical vector is covariant or contravariant. Thus if we define force as ``mass $×$ acceleration,'' the force turns out to be contravariant; but if we define it by ``work $=$~force $×$ displacement,'' the force is covariant. With the latter definition, however, we have to abandon the method of resolution into \emph{oblique} components adopted in elementary mechanics. In what follows it is generally sufficient to confine attention to the mathematical notion of a vector. Some idea of the physical notion will probably give greater insight, but is not necessary for the formal proofs. \Section{22.}{The summation convention} \index{Summation convention}% We shall adopt the convention that whenever a literal suffix appears twice in a term that term is to be summed for values of the suffix $1$, $2$, $3$,~$4$. For example, \Eq{(2.1)}~will be written \[ ds^{2} = g_{\mu\nu}\, dx_{\mu}\, dx_{\nu}\qquad (g_{\nu\mu} = g_{\mu\nu}). \Tag{(22.1)} \] Here, since $\mu$ and $\nu$ each appear twice, the summation \[ \sum_{\mu=1}^{4} \sum_{\nu=1}^{4} \] is indicated; and the result written out in full gives~\Eq{(2.1)}. Again, in the equation \[ A_{\mu}' = \frac{\dd x_{\alpha}}{\dd x_{\mu}'}\, A_{\alpha}, \] the summation on the right is with respect to $\alpha$~only ($\mu$~appearing only once). The equation is equivalent to~\Eq{(19.2)}. The convention is not merely an abbreviation but an immense aid to the analysis, giving it an impetus which is nearly always in a profitable direction. Summations occur in our investigations without waiting for our tardy approval. \PageSep{51} A useful rule may be noted---