If a vector~$A_{\mu}$ receives an infinitesimal increment~$dA_{\mu}$ perpendicular to itself, its length is unaltered to the first order; for by~\Eq{(26.5)} \begin{align*} (l + dl)^{2} &= (A_{\mu} + dA_{\mu}) (A^{\mu} + dA^{\mu}) \\ &= A_{\mu} A^{\mu} + A^{\mu}\, dA_{\mu} + A_{\mu}\, dA^{\mu}\quad\text{to the first order} \\ &= l^{2} + 2A_{\mu}\, dA^{\mu} \quad\text{by~\Eq{(26.3)},} \end{align*} and $A_{\mu}\, dA^{\mu} = 0$ by the condition of perpendicularity~\Eq{(26.4)}. In the elementary vector theory, the scalar-product of two vectors is equal to the product of their lengths multiplied by the cosine of the angle \index{Angle between two vectors}% between them. Accordingly in the general theory the angle~$\theta$ between two vectors $A_{\mu}$ and $B_{\mu}$ is defined by \[ \cos\theta = \frac{A_{\mu} B^{\mu}}{\sqrt{(A_{\alpha} A^{\alpha}) (B_{\beta} B^{\beta})}}. \Tag{(26.6)} \] Clearly the angle so defined is an invariant, and agrees with the usual \index{Invariant!formation of}% definition when the coordinates are rectangular. In determining the angle between two intersecting lines it makes no difference whether the world is curved or flat, since only the initial directions are concerned and these in any case lie in the tangent plane. The angle~$\theta$ (if it is real) has thus the usual geometrical meaning even in non-Euclidean space. It must not, however, be inferred that ordinary angles are invariant for the Lorentz transformation; naturally an angle in three dimensions is invariant only for transformations in three dimensions, and the angle which is invariant for Lorentz transformations is a four-dimensional angle. From a tensor of even rank we can construct an invariant by bringing half the suffixes to the upper and half to the lower position and contracting. Thus from~$A{\mu\nu\sigma\tau}$ we form~$A_{\mu\nu}^{\sigma\tau}$ and contract, obtaining $A = A_{\mu\nu}^{\mu\nu}$. This invariant will be called the \emph{spur}\footnotemark.\footnotetext {Originally the German word \Foreign{Spur}.} Another invariant is the square of the \index{Spur}% length $A_{\mu\nu\sigma\tau} A^{\mu\nu\sigma\tau}$. There may also be intermediate invariants such as $A_{\mu\nu\alpha}^{\alpha} A_{\beta}^{\mu\nu\beta}$. \Section{27.}{Christoffel's $3$-index symbols} \index{Christoffel's $3$-index symbols}% \index{Three-index symbol}%