well as to find the relation between the deformation vector
${\bf u = r - r}_0$ and the discrete coordinates $N^\alf$ introduced, let us
note that the quantities $\p u_i/\p x_k$ obviously coincide with the matrix
elements $\alpha_{ik}$ which describe the coordinate transformations
\begin{equation}
x_i = \,\stackrel{\circ}{x}_i + \, \alpha_{ik}\stackrel{\circ}{x}_k
\ee             %17
and, consequently, the lattice vector transformations:
\begin{equation}
a_{\alf i} = \,\stackrel{\circ}{a}_{\alf i} +
\alpha_{ik}\,\stackrel{\circ}{a}_{\alf k}   .
\ee             %18
    If the deformations are small, then,
\begin{equation}
\ba_\alf = \,\stackrel{\circ}{\ba}_\alf + \delta \ba_\alpha, \qquad
\ba^\beta = \,\stackrel{\circ}{\ba}{\!}^\beta + \delta \ba^\beta.
\ee             %19

    Multiplying these two equations and taking into account relations (7)
and (17)
yield in a linear approximation with respect to $\delta {\bf a}^\beta$
\begin{equation}
\delta a_{\alf i} = -(\,\stackrel{\circ}{a}_{\beta i} \delta a^\beta_k)
\quad \stackrel{\circ}{a}_{\alf k} = \alpha_{ik}\stackrel{\circ}a_{\alpha k} .
\ee         %20
\par
    On the other hand the change of the discrete coordinates owing to the
deformation can be written in the form:
\begin{equation}
N^\alf = N^\alf_0 - w^\alf .
\ee             %21
   where $w^\alpha$ is the deviation from the value $N^\alpha_0$ in the ideal
undeformed lattice.
Taking into account (8) one obtains
\begin{equation}
   \ba^\alf = \nabla N^\alf = \,\stackrel{\circ}{\ba}\!{}\!^\alf - \nabla
w^\alf \ee             %22
    and hence,
\begin{equation}
\delta a^{\alf}_{i} 