Every subinterval provides in addition the \( 2n \) conditions $$ y_i = s(x_i), $$ and $$ s(x_{i+1})= y_{i+1}, $$ to be fulfilled. If we also assume that \( s' \) and \( s'' \) are continuous, then $$ s'_{i-1}(x_i)= s'_i(x_i), $$ yields \( n-1 \) conditions. Similarly, $$ s''_{i-1}(x_i)= s''_i(x_i), $$ results in additional \( n-1 \) conditions. In total we have \( 4n \) coefficients and \( 4n-2 \) equations to determine them, leaving us with \( 2 \) degrees of freedom to be determined.