Appendix 1
From Fig. 3.1 it follows that the components of the vectors \({\varvec{\tau}}\left\{ \tau _i \right\} ,\;{\varvec{\upxi}}\left\{ \xi _i \right\} ,\) and \({\mathbf{k}}\left\{k_i \right\},\,{\mathbf{s}}\left\{s_i \right\}\) are connected with each other by the relationships
$$ \tau_i=k_i\cos\varphi+s_i\sin\varphi , $$
(3.136)
$$ \xi_i=-k_i\sin\varphi+s_i\cos\varphi . $$
(3.137)
Substituting (3.136) and (3.137) into Frenet formulas
$$ {\frac{{\hbox{d}}\tau_i}{{\hbox{d}} s}}=-\tau\xi_,. $$
(3.138)
$$ {\frac{{\hbox{d}} \xi_i}{{\hbox{d}} s}}=\tau\tau_i-\ae\lambda_i , $$
(3.139)
$$ {\frac{{\hbox{d}} \lambda_i}{{\hbox{d}} s}}=\ae\xi_i , $$
(3.140)
we obtain
$$ -\;{\frac{{\hbox{d}} k_i}{{\hbox{d}} s}}\sin\varphi+{\frac{{\hbox{d}} s_i}{{\hbox{d}} s}}\cos\varphi= (K+\tau)k_i\cos\varphi+(K+\tau)s_i\sin\varphi-\ae\lambda_i , $$
(3.141)
$$ {\frac{{\hbox{d}} k_i}{{\hbox{d}} s}}\cos\varphi+{\frac{{\hbox{d}} s_i}{{\hbox{d}} s}}\sin\varphi= (K+\tau)k_i\sin\varphi-(K+\tau)s_i\cos\varphi , $$
(3.142)
$$ {\frac{{\hbox{d}} \lambda_i}{{\hbox{d}} s}}=-k_i\ae\sin\varphi+s_i\ae\cos\varphi . $$
(3.143)
Multiplying (3.141) and (3.142) by \(\cos\varphi\) and \(\sin\varphi,\) respectively, and adding the resulting equations yields
$$ {\frac{{\hbox{d}} s_i}{{\hbox{d}} s}}=(K+\tau)k_i-\ae\lambda_i \cos\varphi . $$
(3.144)
Multiplying further Eqs. 3.141 and 3.142 by \(-\sin\varphi\) and \(\cos\varphi,\) respectively, and adding the resulting equations yields
$$ {\frac{{\hbox{d}} k_i}{{\hbox{d}} s}}=-(K+\tau)s_i+\ae\lambda_i \sin\varphi . $$
(3.145)
Formulas (3.144), (3.145) and (3.143) coincide, respectively, with Eqs. 3.15–3.17.
Appendix 2
$$ \begin{aligned} f_{i(k - 1)} \lambda _i &= {\frac{{{\hbox{d}} ^2 \omega _{(k - 1)} }}{{{\hbox{d}} s^2 }}} - \;{\frac{{\hbox{d}} }{{\hbox{d}} s}} \Bigl( \ae\eta _{(k - 1)} \cos \varphi - \ae\theta _{(k - 1)} \sin \varphi \Bigr) \\ &\quad +\ae\sin\varphi \left\{ {\frac{{\hbox{d}}\theta _{(k-1)}}{{\hbox{d}} s}}+\eta _{(k-1)}(K+\tau ) -\omega _{(k-1)}\ae\sin\varphi \right\} \\ &\quad -\ae\cos\varphi \left\{ {\frac{{\hbox{d}}\eta _{(k-1)}}{{\hbox{d}} s}}-\theta _{(k-1)}(K+\tau ) +\omega _{(k-1)}\ae\cos\varphi \right\} \\ &\quad +\widetilde \omega ^{1y}_{(k)}\ae\sin\varphi -\widetilde \omega ^{1y}_{(k)}\ae\cos\varphi , \end{aligned} $$
(3.146)
$$ \begin{aligned} f_{i(k - 1)} k_i &= {\frac{{{\hbox{d}} ^2 \theta _{(k - 1)} }}{{{\hbox{d}} s^2 }}} + (K + \tau )\left\{ {{\frac{{{\hbox{d}} \eta _{(k - 1)} }}{{{\hbox{d}} s}}} - \theta _{(k - 1)} (K + \tau ) + \omega _{(k - 1)} \ae \cos \varphi } \right\} \\ &\quad - \ae \sin \varphi \left\{ {{\frac{{{\hbox{d}} \omega _{(k - 1)} }}{{{\hbox{d}} s}}} + \theta _{(k - 1)} \ae \sin \varphi - \eta _{(k - 1)} \ae \cos \varphi } \right\} \\ &\quad - {\frac{{\hbox{d}} }{{{\hbox{d}} s}}}\Bigl( \omega _{(k - 1)} \ae \sin \varphi -\eta _{(k - 1)} (K + \tau ) \Bigr) + \omega _{(k - 1)}^{1\lambda } \ae \cos \varphi ,\; \end{aligned} $$
(3.147)
$$ \begin{aligned} f_{i(k - 1)} s_i &= {\frac{{{\hbox{d}} ^2 \eta _{(k - 1)} }}{{{\hbox{d}} s^2 }}} - (K + \tau )\left\{ {{\frac{{{\hbox{d}} \theta _{(k - 1)} }}{{{\hbox{d}} s}}} + \eta _{(k - 1)} (K + \tau ) - \omega _{(k - 1)} \ae \sin \varphi } \right\} \\ &\quad + \ae \cos \varphi \left\{ {{\frac{{{\hbox{d}} \omega _{(k - 1)} }}{{{\hbox{d}} s}}} + \theta _{(k - 1)} \ae \sin \varphi - \eta _{(k - 1)} \ae \cos \varphi } \right\} \\ &\quad - {\frac{{\hbox{d}} }{{{\hbox{d}} s}}}\Bigl( \theta _{(k - 1)} (K + \tau ) - \omega _{(k - 1)} \ae \cos \varphi \Bigr) + \omega _{(k - 1)}^{1\lambda } \ae \sin \varphi , \end{aligned} $$
(3.148)
$$F_{1(k - 1)} = \rho G_1^2 {\frac{{\hbox{d}} }{{{\hbox{d}} s}}}\left\{ {{\frac{{{\hbox{d}} \omega _{(k - 1)}^0 }}{{{\hbox{d}} s}}} - \ae \left(\eta _{(k - 1)}^0 \cos \varphi - \theta _{(k - 1)}^0 \sin \varphi \right) }+ \ae \left(a_x \cos \varphi + a_y \sin \varphi \right)\omega _{(k - 1)}^{1\lambda } \right\} + 2\ae \rho G_2^2 \left\{ \left(a_x \cos \varphi + a_y \sin \varphi \right) {\frac{{{\hbox{d}} \omega _{(k - 1)}^{1\lambda } }}{{{\hbox{d}} s}}} + (K + \tau )\left(a_y \cos \varphi - a_x \sin \varphi \right)\omega _{(k - 1)}^{1\lambda } \right\} + 2\rho G_2^2 (K + \tau )\omega _{(k - 1)}^{1\lambda } +2\ae \rho G_2^2\left\{\cos\varphi \left( \omega _{(k - 1)}^{1 x} -{\frac{{\hbox{d}} \eta _{(k - 1)}^0 }{{\hbox{d}} s}} \right) -\sin\varphi \left( \omega _{(k - 1)}^{1 y}- {\frac{{\hbox{d}} \theta _{(k - 1)}^0 }{{\hbox{d}} s}} \right)\right\} + \rho G_2^2 \ae \left\{- \omega _{(k - 1)}^{1x} \cos \varphi + \omega _{(k - 1)}^{1y} \sin \varphi -\ae \omega _{(k - 1)}^0 \right. +\left. (K + \tau ) \left( \theta _{(k - 1)}^0\cos\varphi + \eta _{(k - 1)}^0\sin\varphi \right) \right\} + f_{i(k - 1)}^0 \lambda _i \sigma _{\lambda \lambda }^0 ,$$
(3.149)
$$ \begin{aligned} F_{2(k - 1)} &= \rho G_2^2 {\frac{{\hbox{d}} }{{{\hbox{d}} s}}}\left\{ {\frac{{{\hbox{d}} \theta _{(k - 1)}^0 }}{{{\hbox{d}} s}}} - \omega _{(k - 1)}^{1y} + {\frac{{{\hbox{d}} \omega _{(k - 1)}^{1\lambda } }}{{{\hbox{d}} s}}}a_y - (K + \tau )a_x \omega _{(k - 1)}^{1\lambda } \right\} \\ &\quad + \rho G_2^2 (K + \tau )\left\{ {{\frac{{{\hbox{d}} \eta _{(k - 1)}^0 }}{{{\hbox{d}} s}}} - \omega _{(k - 1)}^{1x} - {\frac{{{\hbox{d}} \omega _{(k - 1)}^{1\lambda } }}{{{\hbox{d}} s}}}a_x - (K + \tau )a_y \omega _{(k - 1)}^{1\lambda } } \right\} \\ &\quad - \rho G_1^2 \ae \left\{ {{\frac{{{\hbox{d}} \omega _{(k - 1)}^0 }}{{{\hbox{d}} s}}} - \ae \left(\eta _{(k - 1)}^0 \cos \varphi - \theta _{(k - 1)}^0 \sin \varphi \right) } \right. \\ &\quad + \left. \ae \left(a_x \cos \varphi + a_y \sin \varphi \right) \omega _{(k - 1)}^{1\lambda } \right\}\sin \varphi + f_{i(k - 1)}^0 k_i \sigma _{\lambda \lambda }^0 , \end{aligned} $$
(3.150)
$$ \begin{aligned} F_{3(k - 1)} &= \rho G_2^2 {\frac{{\hbox{d}} }{{{\hbox{d}} s}}}\left\{ {{\frac{{{\hbox{d}} \eta _{(k - 1)}^0 }}{{{\hbox{d}} s}}} - \omega _{(k - 1)}^{1x} - {\frac{{{\hbox{d}} \omega _{(k - 1)}^{1\lambda } }}{{{\hbox{d}} s}}}a_x - (K + \tau )a_y \omega _{(k - 1)}^{1\lambda } } \right\} \\ &\quad - \rho G_2^2 (K + \tau )\left\{ {\frac{{{\hbox{d}} \theta _{(k - 1)}^0 }}{{{\hbox{d}} s}}} - \omega _{(k - 1)}^{1y} + {\frac{{{\hbox{d}} \omega _{(k - 1)}^{1\lambda } }}{{{\hbox{d}} s}}}a_y - (K + \tau )a_x \omega _{(k - 1)}^{1\lambda } \right\} \\ &\quad + \rho G_1^2 \ae \left\{ {\frac{{{\hbox{d}} \omega _{(k - 1)}^0 }}{{{\hbox{d}} s}}} - \ae \left(\eta _{(k - 1)}^0 \cos \varphi - \theta _{(k - 1)}^0 \sin \varphi \right) \right. \\ &\quad + \left. \ae \left(a_x \cos \varphi + a_y \sin \varphi \right)\omega _{(k - 1)}^{1\lambda } \right\}\cos \varphi + f_{i(k - 1)}^0 s_i \sigma _{\lambda \lambda }^0 , \end{aligned} $$
(3.151)
$$ \begin{aligned} F_{4(k - 1)} &= \rho G_1^2 {\frac{{\hbox{d}}^2 }{{\hbox{d}} s^2 }}\left[ -\omega _{( k - 1)}^{1x} -\omega _{( k - 1)}^0\ae\cos\varphi+(K+\tau )\theta _{( k - 1)}^0\right] \\ &\quad -2\ae^2 \rho G_2^2\left[- \omega _{(k - 1)}^{1x} -\omega _{(k - 1)}^0\ae\cos\varphi +\theta _{(k - 1)}^0 (K + \tau ) \right] \\ &\quad - 2\ae \rho G_2^2 \left[ {\frac{{\hbox{d}} \omega _{(k - 1)}^{1\lambda } }{{\hbox{d}} s}}\sin \varphi +(K + \tau )\omega _{( k - 1)}^{1\lambda } \cos \varphi \right] \\ &\quad -\rho G_1^2 {\frac{{\hbox{d}} }{{\hbox{d}} s}}\left( \ae\sin\varphi \omega _{(k - 1)}^{1\lambda }\right) + I_x^{ - 1} \sigma _{\lambda \lambda }^0 \int\limits_F {f_{i(k - 1)} \lambda _i y \;{\hbox{d}} F} , \end{aligned} $$
(3.152)
$$ \begin{aligned} F_{5(k - 1)} &= \rho G_1^2 {\frac{{\hbox{d}}^2 }{{\hbox{d}} s^2}}\left[ -\omega _{( k - 1)}^{1y} +\omega _{( k - 1)}^0\ae\sin\varphi-(K+\tau )\eta _{( k - 1)}^0\right] \\ &\quad -2\ae^2 \rho G_2^2\left[- \omega _{(k - 1)}^{1y} +\omega _{(k - 1)}^0\ae\sin\varphi -\eta _{(k - 1)}^0(K + \tau ) \right] \\ &\quad - 2\ae \rho G_2^2 \left[ {\frac{{\hbox{d}} \omega _{(k - 1)}^{1\lambda } }{{\hbox{d}} s}}\cos \varphi - (K + \tau )\omega _{( k - 1)}^{1\lambda } \sin \varphi \right] \\ &\quad -\rho G_1^2 {\frac{{\hbox{d}} }{{\hbox{d}} s}}\left( \ae\cos\varphi \omega _{(k - 1)}^{1\lambda }\right) + I_y^{ - 1} \sigma _{\lambda \lambda }^0 \int\limits_F {f_{i(k - 1)} \lambda _i x \;{\hbox{d}} F} , \end{aligned} $$
(3.153)
$$F_{6 (k - 1)} = - \ae ^2 (2\rho G_2^2+\sigma _{\lambda \lambda }^0) \Bigl\{- \psi _{(k - 1)} + \ae \cos \varphi \left[ -\omega _{( k - 1)}^{1y} +\omega _{( k - 1)}^0\ae\sin\varphi-(K+\tau )\eta _{( k - 1)}^0\right] - \left.\ae \sin \varphi \left[ -\omega _{( k - 1)}^{1x} -\omega _{( k - 1)}^0\ae\cos\varphi+(K+\tau )\theta _{( k - 1)}^0\right] \right\} +\left( \rho G_1^2 + \sigma _{\lambda \lambda }^0 \right) {\frac{{\hbox{d}}^2 }{{\hbox{d}} s^2}} \Bigl\{ - \psi _{(k - 1)} + \ae \cos \varphi \left[ -\omega _{( k - 1)}^{1y} +\omega _{( k - 1)}^0\ae\sin\varphi-(K+\tau )\eta _{( k - 1)}^0\right] -\left. \ae \sin \varphi \left[ -\omega _{( k - 1)}^{1x} -\omega _{( k - 1)}^0\ae\cos\varphi+(K+\tau )\theta _{( k - 1)}^0\right] \right\},$$
(3.154)
$$F_{7(k - 1)} = \rho G_2^2 {\frac{{\hbox{d}} }{{\hbox{d}} s}} \left\{ I_p^A {\frac{{\hbox{d}} \omega _{(k - 1)}^{1\lambda }}{{\hbox{d}} s}} \right. +\ae I_y \cos \varphi \left[ -\omega _{( k - 1)}^{1y} +\omega _{( k - 1)}^0\ae\sin\varphi-(K+\tau )\eta _{( k - 1)}^0\right] +\ae I_x \sin \varphi \left[ -\omega _{( k - 1)}^{1x} -\omega _{( k - 1)}^0\ae\cos\varphi+(K+\tau )\theta _{( k - 1)}^0\right] + \left. a_x F\left(\omega_{(k-1)}^{1x}-{\frac{{\hbox{d}} \eta _{( k - 1)}^0 }{{\hbox{d}} s}}\right) - a_y F\left(\omega_{(k-1)}^{1y}-{\frac{{\hbox{d}} \theta _{( k - 1)}^0 }{{\hbox{d}} s}} \right)\right\} -\rho G_2^2 F(K+\tau )\left\{ a_y \left(\omega_{(k-1)}^{1x}-{\frac{{\hbox{d}} \eta _{( k - 1)}^0 }{{\hbox{d}} s}}\right) + a_x \left(\omega_{(k-1)}^{1y}-{\frac{{\hbox{d}} \theta _{( k - 1)}^0 }{{\hbox{d}} s}} \right)\right\} +\rho G_2^2 (K+\tau )\ae I_y\sin\varphi \left[ -\omega _{( k - 1)}^{1y}+\omega _{( k - 1)}^0\ae\sin\varphi-(K+\tau )\eta _{( k - 1)}^0\right] -\rho G_2^2 (K+\tau )\ae I_x\cos\varphi \left[ -\omega _{( k - 1)}^{1x}-\omega _{( k - 1)}^0\ae\cos\varphi+(K+\tau )\theta _{( k - 1)}^0\right] +\rho G_1^2 \ae F\left(a_x\cos\varphi +a_y\sin\varphi \right) \left[- {\frac{{\hbox{d}} \omega _{(k - 1)}^0 }{{\hbox{d}} s}} +\ae\left(\eta _{( k - 1)}^0 \cos\varphi -\theta _{( k - 1)}^0 \sin\varphi \right) -\ae(a_x\cos\varphi +a_y\sin\varphi)\omega _{( k - 1)}^{1\lambda }\right] + \rho G_1^2 \ae \left\{ I_y\cos\varphi {\frac{{\hbox{d}} }{{\hbox{d}} s}} \left[ -\omega _{( k - 1)}^{1y}+\omega _{( k - 1)}^0\ae\sin\varphi-(K+\tau )\eta _{( k - 1)}^0\right] \right. +I_x \sin\varphi {\frac{{\hbox{d}} }{{\hbox{d}} s}} \left[ -\omega _{( k - 1)}^{1x}-\omega _{( k - 1)}^0\ae\cos\varphi+(K+\tau )\theta _{( k - 1)}^0\right] - \left. \ae\left(I_y\cos^2\varphi +I_x\sin^2\varphi \right)\omega _{( k - 1)}^{1\lambda }\right\} -\rho G_2^2 F(K+\tau )^2 I_p^A \omega _{(k - 1)}^{1\lambda } +\sigma _{\lambda \lambda }^0 \int\limits_F^{} {\left[ f_{i( k - 1)} s_i ( x - a_x ) -f_{i( k - 1)} k_i ( y - a_y ) \right] {\hbox{d}} F.}$$
(3.155)
Appendix 3
$$ \begin{aligned} A_{0(k)}(s)&=\left\{ G_I^{ - 1} \left( \rho G_1^2 + 2\rho G_2^2 + 2\sigma _{\lambda \lambda }^0 \right)\ae \left[ -\left(a_x \cos \varphi + a_y \sin \varphi \right)\omega _{( k )}^{1\lambda } \right.\right. \\ &\quad + \left.\left. \eta _{( k )}^0 \cos \varphi - \theta _{( k )}^0 \sin \varphi \right] + F_{1(k - 1)} \left| {_{G = G_I } } \right. \right\} {\frac{1}{2\;\rho G_I }}, \end{aligned} $$
(3.156)
$$ \begin{aligned} A_{1(k)}(s)&=\left\{ {2\;\rho G_I } {\frac{{\hbox{d}}}{{\hbox{d}} s}} \left[ \theta _{( k )}^0 (K+\tau )\right] \right. -\ae (\rho G_1^2 + 2\rho G_2^2 + 2\sigma _{\lambda \lambda }^0 ) G_I^{ - 1} \omega _{( k )}^{1\lambda } \sin \varphi \\ &\quad - F_{4(k - 1)} \left| {_{G = G_I } } \right. \Bigr\} {\frac{1}{2\;\rho G_I }}, \end{aligned} $$
(3.157)
$$ \begin{aligned} A_{2(k)}(s)&=\left\{ -2 \rho G_I {\frac{{\hbox{d}}}{{\hbox{d}} s}} \left[\eta _{( k )}^0 (K+\tau )\right] \right. -\ae (\rho G_1^2 + 2\rho G_2^2 + 2\sigma _{\lambda \lambda }^0 ) G_I^{ - 1} \omega _{( k)}^{1\lambda } \cos \varphi \\ &\quad - F_{5(k - 1)} \left| {_{G = G_I } } \right. \Bigr \} {\frac{1}{2\;\rho G_I }}, &(158) \\ \end{aligned} $$
(3.158)
$$ \begin{aligned} A_{3(k)}(s)&=\left\{ -2 \rho G_I {\frac{{\hbox{d}}}{{\hbox{d}} s}} \left[\left(\theta _{( k )}^0 \sin\varphi + \eta _{( k )}^0 \cos\varphi \right) \ae (K+\tau ) \right]\right.\\ &\quad - F_{6(k - 1)} \left| {_{G = G_I } } \right. \Bigr \} {\frac{1}{2\;\rho G_I }}, \end{aligned} $$
(3.159)
$$ \begin{aligned} B_{1(k-1)}&= -\omega _{( k )}^0\ae\cos\varphi -\Bigl\{ 2 G_{II}^{ - 1} \left( {\rho G_1^2 + \sigma _{\lambda \lambda }^0 } \right) \\ &\quad \times {\frac{{\hbox{d}}}{{\hbox{d}} s}}\left[ \omega _{( k - 1)}^{1x} + \omega _{( k-1)}^0 \ae \cos \varphi -\theta_{( k-1)}^0(K+\tau ) \right] \\ &\quad + \ae \left( {\rho G_1^2 + 2\rho G_2^2 + 2\sigma _{\lambda \lambda }^0 } \right)G_{II}^{ - 1} \omega _{( k - 1)}^{1\lambda } \sin \varphi \\ &\quad + F_{4(k - 2)} \left| {_{G = G_{II} } } \right. \Bigr\} G_{II}^2\rho ^{-1} \left( G_2^2 - G_1^2 \right)^{-1}, \end{aligned} $$
(3.160)
$$ \begin{aligned} B_{2(k-1)}&= \omega _{( k )}^0\ae\sin\varphi -\Bigl\{ 2G_{II}^{ - 1} \left( {\rho G_1^2 + \sigma _{\lambda \lambda }^0 } \right) \\ &\quad \times {\frac{{\hbox{d}}}{{\hbox{d}} s}}\left[ \omega _{( k - 1)}^{1y} - \omega _{( k-1)}^0 \ae \sin \varphi + \eta _{( k-1)}^0(K+\tau )\right] \\ &\quad + \ae \left( {\rho G_1^2 + 2\rho G_2^2 + 2\sigma _{\lambda \lambda }^0 } \right)G_{II}^{ - 1} \omega _{( k - 1)}^{1\lambda } \cos \varphi \\ &\quad + F_{5( k - 2)} \left| {_{G = G_{II} } } \right.\Bigr \} G_{II}^2\rho ^{-1} \left( G_2^2 - G_1^2 \right)^{-1}, \end{aligned} $$
(3.161)
$$ \begin{aligned} B_{3(k)}&=\left\{ G_{II}^{ - 1} \ae \left( {\rho G_1^2 + \rho G_2^2 + 2\sigma _{\lambda \lambda }^0 } \right)\omega _{( k )}^0 \sin \varphi \right. + G_{II}^{ - 1} \rho G_2^2 \left( B_{2(k-1)} -\ae \omega _{( k )}^0 \sin \varphi\right) \\ &\quad + F_{2( k - 1)} \left| {_{G = G_{II} } } \right. \Bigr \}\frac{1}{2}\;G_{II} \left( \rho G_2^2 + \sigma _{\lambda \lambda }^0 \right)^{-1}, \end{aligned} $$
(3.162)
$$B_{4(k)}=\left\{ - G_{II}^{ - 1} \ae \left( {\rho G_1^2 + \rho G_2^2 + 2\sigma _{\lambda \lambda }^0 } \right)\omega _{( k )}^0 \cos \varphi \right. + G_{II}^{ - 1} \rho G_2^2 \left( B_{1(k-1)} +\ae \omega _{( k )}^0 \cos \varphi\right) + F_{3( k - 1)} \left| {_{G = G_{II} } } \right. \Bigr \}\frac{1}{2}\;G_{II} \left( \rho G_2^2 + \sigma _{\lambda \lambda }^0 \right)^{-1},$$
(3.163)
$$ \begin{aligned} B_{5(k)}&=\left\{ G_{II}^{ - 1} \ae \left( {\rho G_1^2 + \rho G_2^2 + 2\sigma _{\lambda \lambda }^0 } \right) \left[ B_{1(k-1)}I_x \sin \varphi+B_{2(k-1)}I_y \cos \varphi \right. \right. \\ &\quad +\left. \omega _{( k )}^0 F\left( a_x \cos \varphi + a_y \sin \varphi \right) +\frac{1}{2}\;\omega _{( k )}^0\ae(I_x-I_y)\sin2\varphi \right] \\ &\quad + G_{II}^{ - 1} \rho G_2^2 F\left[ B_{2(k-1)}a_y -B_{1(k-1)} a_x -\omega _{\left( k \right)}^0 \ae\left( a_x \cos \varphi + a_y \sin \varphi \right) \right] \\ &+ F_{7( k - 1)} \left|_{G = G_{II} }, \right. \Bigr\} \frac{1}{2}\;G_{II} \left( \rho G_2^2 + \sigma _{\lambda \lambda }^0 \right)^{-1}. \end{aligned} $$
(3.164)
