properties of third order derivative

I have a question in my assignment (about interpolation) which has the condition that the third-order derivative is continuous at $x_2$ and $x_{N-1}$. That is, $S'''(x_2)=S'''(x_{N-2})$.

The question is about interpolation of N points using cubic spline interpolant.

I'm wondering how I can use the third-derivative condition. What kind of properties does it have?

The condition you wrote down is not quite correct. If the spline is third order continuous at $x=x_2$, then this means that the first and second segments have the same third derivative at this point. So, if $s_i$ denotes the $i$-th segment, we are told that $s_1'''(x_2) = s_2'''(x_2)$ and $s_{N-3}'''(x_{N-2}) = s_{N-2}'''(x_{N-2})$. We don't know that $s'''(x_2) = s'''(x_{N-2})$.
The exact nature of the two extra equations depends on how you are representing the cubic segments. If each of them is written in the form $s_i(x) = a_ix^3 +b_ix^2 + c_ix = d_i$, for example, then the condition $s_1'''(x_2) = s_2'''(x_2)$ tells you that $a_1 = a_2$.