this we have to calculate the dispersion of the vector ${\bf
m}(\mbox{\boldmath$\varrho$})$ with respect to the transverse coordinate $\mbox{\boldmath$\varrho$}$: $$\begin{aligned}
&& <m_i m_j> - <m_i>< m_j>= \int m_i({\bf r}_\perp -{\bf x})
m_j({\bf r}_\perp -{\bf x}) w_c({\bf x})d^2x
\nonumber \\
&& - \int m_i({\bf r}_\perp -{\bf x})w_c({\bf x})d^2x \int
m_j({\bf r}_\perp -{\bf x})w_c({\bf x})d^2x,
\label{3.6}\end{aligned}$$ where $w_c({\bf x})$ is the distribution function of target particles normalized to the unity.

As a result we obtain the following expression for the correction to the probability of photon emission connected with the restricted transverse dimensions of colliding beams of charged particles: $$\begin{aligned}
&& dw_1=\frac{\alpha}{(2\pi)^2}\frac{d^3k}{\omega}T_{ij}({\bf e},
{\bf p}_{\perp}, s, s')L_{ij}, \quad
T_{ij}=\left[\frac{\partial}{\partial p_{\perp
i}}\frac{R^{\ast}({\bf
p}_{\perp})}{k'v}\right]\left[\frac{\partial}{\partial p_{\perp
j}}\frac{R^{\ast}({\bf p}_{\perp})}{k'v}\right],
\nonumber \\
&& L_{ij}=\int m_i(\mbox{\boldmath$\varrho$})
m_j(\mbox{\boldmath$\varrho$})\left(w_r({\bf
x}+\mbox{\boldmath$\varrho$})-w_r({\bf x})\right)w_c({\bf
x})d^2xd^2\varrho
\nonumber \\
&&- \left(\int m_i(\mbox{\boldmath$\varrho$})w_c({\bf
x}-\mbox{\boldmath$\varrho$})d^2\varrho \right)\left(\int
m_j(\mbox{\boldmath$\varrho$})w_c({\bf
x}-\mbox{\boldmath$\varrho$})d^2\varrho \right)w_r({\bf x })d^2x.
\label{3.7}\end{aligned}$$

Averaging over the polarization of initial electrons and summing over the polarization of final electrons we find $$T_{ij}=\frac{l_f}{\varepsilon\varepsilon'}\Big[e_i e_j-
\frac{2{\bf e u}}{\zeta}(e_i u_j + u_i e_j) + \frac{4({\bf e
u})^2}{\zeta^2} u_i u_j+
\frac{\omega^2}{4\varepsilon\varepsilon'}\delta_{ij}\Big].
\label{3.8}$$ Note, that one can choose the real vector [**e**]{} since only the linear polarization could arise in the case of unpolarized electrons.

After summation in (\[3.8\]) over the polarization of emitted 