that ∂W$_{i}$ ( y ) ∂y 0 1 = τ ( y ) ∫ θ$_{i}$ ∂f ( y ( s )) ∂y ∂y ( s ) ∂y 0 1 ds + f ( y ( s )) ∂τ ( y ) ∂y 0 1 + J$_{y}$ ( y )( e$_{1}$ + f ( y ( s )) ∂τ ( y ) ∂y 0 1 ) + f ( y ( s ) + J ( y ( s ))) ∂τ ( y ) ∂y 0 1 + θ$_{i}$ ∫ τ ( y ) ∂f ( y ( s ) + J ( y ( s ))) ∂x ∂y ( s ) ∂y 0 1 ds, (3.17) ∂W$_{i}$ ( y ( ξ, 0 , y ∗ + Δ y )) ∂y 0 1 = f ( y ( ξ, 0 , y ∗ + Δ y )) ∂τ ( y ( ξ, 0 , y ∗ + Δ y )) ∂y 0 1 + J$_{y}$ ( y ( ξ, 0 , y ∗ + Δ y )) ( e$_{1}$ + f ( y ( ξ, 0 , y ∗ + Δ y ))) ∂τ ( y ( ξ, 0 , y ∗ + Δ y ))) ∂y 0 1 ) + f ( y ( ξ, 0 , ( J ( y ( ξ, 0 , y ∗ + Δ y ))))) ∂τ ( J ( y ( ξ, 0 , y ∗ + Δ y ))) ∂y 0 1 . (3.18) ∂W$_{i}$ (¯ y ) ∂y 0 1 =    ¯ y$_{2}$ − ¯ y$_{1}$ − 0 . 001¯ $_{y$_{2}$}$   ( − 1 ¯ y$_{2}$ ) +    1 0 0 − 2 R$_{1}$ ¯ $_{y$_{2}$}$   ( e$_{1}$ +    ¯ y$_{2}$ − ¯ y$_{1}$ − 0 . 001¯ $_{y$_{2}$}$   ( − 1 ¯ y$_{2}$ ) (3.19) −    − R$_{1}$ (¯ y$_{2}$ ) 2 − ¯ y$_{1}$ + 0 . 001 R$_{1}$ (¯ y$_{2}$ ) $_{2}$   ( − 1 ¯ y$_{2}$ ) =    ¯ y$_{2}$ − R$_{1}$ (¯ y$_{2}$ ) 2 − ¯ y$_{1}$ − 0 . 001(¯ y$_{2}$ − R$_{1}$ (¯ y$_{2}$ ) 2 $_{)}$   ( − 1 ¯ y$_{2}$ ) +    1 0 0 − 2 R$_{1}$ ¯ $_{y$_{2}$}$      0 ¯ y$_{1}$ + 0 . 001¯ y$_{2}$ ¯ y$_{2}$    . ∂W$_{i}$ (¯ y ) ∂y 0 1 =    − R$_{1}$ ¯ y$_{2}$ − 1 0 . 001(1 − R$_{1}$ ¯ y$_{2}$ ) + 2 R$_{1}$ (0 . 001¯ y$_{2}$ − ¯ y$_{1}$ $_{)}$   . (3.20) where e$_{1}$ = (1 , 0) $^{T}$, T denotes the transpose of a matrix. Substituting y = ¯ y to the formula (3.17) , we obtain that Considering the formula (3.10) for the transversal point ¯ y = (¯ y$_{1}$, ¯ y$_{2}$ ) , the first component ∂τ (¯ y ) ∂y 0 1 can be evaluated as ∂τ (¯ y ) ∂y 0 1 = − 1 ¯ y$_{2}$ . From the last equality, it is seen how the singularity appears at the grazing point. Finally, we obtain that Calculating the righthand side of (3.19) we have The last expression demonstrates that the derivative is a continuous function of its arguments in a neighborhood of the grazing point. Since it is defined and continuous for the points, which are not from the grazing orbit by the last expression and for other points it can be determined by the limit procedure. 16