The principal operations of the tensor calculus are addition, multiplication (outer and inner), summation (\SecRef{22}), contraction (\SecRef{24}), substitution (\SecRef{25}), raising and lowering suffixes (\SecRef{26}), covariant differentiation (\SecRefs{29}, \SecNum{30}). There is no operation of division; but an inconvenient factor $g_{\mu\nu}$ or~$g^{\mu\nu}$ can be removed by multiplying through by $g^{\mu\sigma}$ or~$g_{\mu\sigma}$ so as to form the substitution-operator. The operation of summation is practically outside our control and always presents itself as a \Foreign{fait accompli}. The most characteristic process of manipulation in this calculus is the free alteration of dummy suffixes (those appearing twice in a term); it is probably this process which presents most difficulty to the beginner. Of special interest are the fundamental tensors or world-tensors, of which we have discovered two, viz.\ $g_{\mu\nu}$ and~$B_{\mu\nu\sigma\rho}$. The latter has been expressed in terms of the former and its first and second derivatives. It is through these that the gap between pure geometry and physics is bridged; in particular $g_{\mu\nu}$~relates the observed quantity~$ds$ to the mathematical coordinate specification~$dx_{\mu}$. Since in our work we generally deal with tensors, the reader may be led to overlook the rarity of this property. The juggling tricks which we seem to perform in our manipulations are only possible because the material used is of quite exceptional character. The further development of the tensor calculus will be resumed in \SecRef{48}; but a stage has now been reached at which we may begin to apply it to the theory of gravitation. \PageSep{76} \Chapter{III}{The Law of Gravitation} \Section[The condition for flat space-time]{36.}{The condition for flat space-time. Natural coordinates} \index{Flat space-time!condition for}% A region of the world is called \emph{flat} or \emph{homaloidal} if it is possible to construct in it a Galilean frame of reference. It was shown in \SecRef{4} that when the $g_{\mu\nu}$ are constants, $ds^{2}$~can be reduced to the sum of four squares, and Galilean coordinates can be constructed. Thus an equivalent definition of flat space-time is that it is such that coordinates can be found for which the $g_{\mu\nu}$ are constants. When the $g_{\mu\nu}$ are constants the $3$-index symbols all vanish; but since the $3$-index symbols do not form a tensor, they will not in general continue to vanish when other coordinates are substituted in the same flat region. Again, when the $g_{\mu\nu}$ are constants, the Riemann-Christoffel tensor, being composed \index{Riemann-Christoffel tensor!vanishing of}% of products and derivatives of the $3$-index symbols, will vanish; and since it is a tensor, it will continue to vanish when any other coordinate-system is substituted in the same region.