diff --git a/Lectures_my/NumMet/Lecture2/mchrzasz.tex b/Lectures_my/NumMet/Lecture2/mchrzasz.tex
index d374721..ceb7088 100644
--- a/Lectures_my/NumMet/Lecture2/mchrzasz.tex
+++ b/Lectures_my/NumMet/Lecture2/mchrzasz.tex
@@ -696,13 +696,63 @@
 p_k&=\frac{y_k}{h_k}-\frac{m_k h_k}{6}\\
 p_{k+1}&=\frac{y_{k+1}}{h_{k+1}}-\frac{m_{k+1} h_{k+1}}{6}
 \end{align*}
+\ARROW Putting all the things together:
+\begin{align}
+S_k(x)= &\frac{m_k}{6h_k} (x_k-x)^3 + \frac{m_{k+1}}{6h_k}(x-x_k)^3+ \left( \frac{y_k}{h_k}  -\frac{m_k h_k}{6}\right) (x_{k+1}-x)\\ + & \left(\frac{y_{k+1}}{h_k} - \frac{m_{k+1} h_k}{6} \right) (x-x_k)
+\label{eq:almostthere}
+\end{align}
+\end{small}
+\end{frame}
+
+
+\begin{frame}\frametitle{Third degree splain function}
+\begin{small}
+\ARROW The only unknown in the above equation are the $m_i$ variables. To get those we need to use the equation: $S^{\prime}_{k-1}(x_k)=s^{\prime}(x_k)$:
+\begin{align*}
+-\frac{1}{3}m_k h_k - \frac{1}{6} m_{k+1}h_k + d_k = \frac{1}{3}m_k h_{k-1} + \frac{1}{6}m_{k-1}h_{k-1}+d_{k-1} 
+\end{align*}
+\ARROW So the complete solution is:
+\begin{align*}
+a_{k,0}&=y_k\\
+a_{k,1}&=d_k -\frac{h_k}{6}(2m_k+m_{k+1})\\
+a_{k,2}&= \frac{m_k}{2}\\
+a_{k,3}&= \frac{m_{k+1}-m_k}{6 h_k}
+\end{align*}
+\ARROW To get the $m_0$ and $m_n$ one needs to use one of the aforementioned conditions.
+
+\end{small}
+\end{frame}
+
+
+\begin{frame}\frametitle{Third degree splain function algorithm}
+\begin{small}
+
+\begin{itemize}
+\item Get the input interpolation points.
+\item Calculate the temporary variables: $d_i, h_i$.
+\item Assume adequate conditions to get $m_0$, $m_n$.
+\item Solve linear equation system.
+\item Obtained values of the $a_{i,j}$ coefficients put in the interpolation equation.
+\end{itemize}
+
 
 \end{small}
 \end{frame}
 
 
 
+\begin{frame}\frametitle{Summary}
+\begin{small}
 
+\ARROW Interpolation playes essential role in almost all numerical methods!\\
+\ARROW Even an very old algorithms like Lagrange are still used(we used it in splains).\\
+\ARROW There is a lot of algorithms on the market. From simple Lagrange and Newton interpolation algorithms up to modern splains.\\
+\ARROW The most common used today are splains. They require a bit of work but they work very effectively.\\
+ 
+
+
+\end{small}
+\end{frame}
 
 
 \backupbegin