SUM of N SLIDERS

Theorem 4.3 cannot be generalized to $N$ sliders $(N\geq 3)$: in other words, the sum of 3 or more sliders can yield a slider without their axes intersecting or being parallel. Here is an example: in $(O, {\bf \hat{e}}_1 , {\bf \hat{e}}_2, {\bf \hat{e}}_3)$ consider the three sliders \[ \{ {\cal S}_1 \} = \begin{Bmatrix}{\bf \hat{e}}_1 \\ {\bf 0} \end{Bmatrix}_{O}, \quad \{ {\cal S}_2 \} = \begin{Bmatrix}{\bf \hat{e}}_2 \\ {\bf 0} \end{Bmatrix}_{A}, \quad \{ {\cal S}_3 \} = \begin{Bmatrix}{\bf \hat{e}}_3 \\ {\bf 0} \end{Bmatrix}_{B} \] with ${\bf r}_{OA}= {\bf \hat{e}}_3$ and ${\bf r}_{OB}= {\bf \hat{e}}_1+ {\bf \hat{e}}_2$. Consider the screw \[ \{ {\cal V} \} = \alpha \{ {\cal S}_1 \}+ \beta \{ {\cal S}_2 \}+\gamma \{ {\cal S}_3 \} \] with $\alpha$, $\beta$, and $\gamma$ non-zero scalars. It is easy to see that the axes of these sliders are neither parallel nor intersecting. Yet it is possible to find a condition on the scalars $(\alpha,\beta,\gamma)$ for which $\{ {\cal V} \}$ is a slider: we impose the pitch of $\{ {\cal V} \}$ to be zero: \[ (\alpha {\bf \hat{e}}_1 + \beta {\bf \hat{e}}_2 + \gamma {\bf \hat{e}}_3 ) \cdot ( \beta {\bf \hat{e}}_2 \times {\bf \hat{e}}_3 + \gamma {\bf \hat{e}}_3 \times ({\bf \hat{e}}_1+ {\bf \hat{e}}_2) ) =0 \] giving the equation \[ \alpha \gamma = \alpha\beta + \beta\gamma \]