Using the conditions $s_i(x_i)=y_i$ and $s_i(x_{i+1})=y_{i+1}$ we can in turn determine the constants $c$ and $d$ resulting in $$\begin{align} s_i(x) =&\frac{f_i}{6(x_{i+1}-x_i)}(x_{i+1}-x)^3+ \frac{f_{i+1}}{6(x_{i+1}-x_i)}(x-x_i)^3 \nonumber \\ +&(\frac{y_{i+1}}{x_{i+1}-x_i}-\frac{f_{i+1}(x_{i+1}-x_i)}{6}) (x-x_i)+ (\frac{y_{i}}{x_{i+1}-x_i}-\frac{f_{i}(x_{i+1}-x_i)}{6}) (x_{i+1}-x). \tag{19} \end{align}$$