$$\begin{aligned}
 \sigma^i(x) &=& \left|{\partial X \over \partial x}
  \right|^{(3)} \biggl\{ v_A^i e^0_{A 0} \Sigma + e^i_{A a} \Sigma^a
  \nonumber \\
  & & \qquad\qquad\; + {1 \over c^2} \Bigl[ 2v_A^i G_{A}^{\prime
  \prime} + c^2
  e^{i (1)}_{A a} X^a - v_A^i A_{A a}X^a + R^i_{A a}\left({1 \over 2}
  A^{(1)}_{A a} X^2 - A^{(1)}_{A b} X^b X^a\right)\Bigr] \Sigma
 \nonumber \\
  & & \qquad\qquad\; +  {1 \over c^2} \Bigl[2R^i_{A a} G_A^{\prime
   \prime}
   + v_A^i V_A^a - R^i_{A a} A^b_{A} X^b + R^i_{A b} \partial_a
  \left({1 \over 2} A^b_{A} X^2 - A^c_{A} X^c X^b\right)\Bigr]
  \Sigma^a \nonumber \\
 & & \qquad\qquad\; - {1 \over c^2}\Bigl[v_A^i \partial_T\left(X^b
  V_A^b \Sigma\right)+ R^i_{A a} \partial_T\left( X^bV_A^b \Sigma^a
  \right)\Bigr] + {1 \over c^2} V_A^a R^i_{A b} \left[{\bf T}^{ab} -
  \delta_{ab} {\bf T}^{cc}\right] \biggr\} \nonumber \\
 & & + {\cal O}(4) \; , \label{twentyfive}\end{aligned}$$ (where $e^{i (1)}_{A a} = de^i_{A a} / dT_A$, etc.) generalizing the formula $\sigma^i = v_A^i \Sigma + R^i_{A a}
\Sigma^a + {\cal O}(2)$ used throughout the series of papers [@dam1; @dam2; @dam3; @dam4] \[Note the three-dimensional Jacobian, as defined in Eq. (\[six\]) above, in front of Eq. (\[twentyfive\])\].

Inserting this relation into the defining integral (\[twentyfour\]) and using the method of Sec.II.A one obtains an expression of 