What is the three-moment equation and when is it used?

The three-moment equation (Clapeyron's theorem) relates moments at three consecutive supports of a continuous beam: M1*L1 + 2*M2*(L1+L2) + M3*L2 = -6*(A1*a1/L1 + A2*b2/L2), where A and a/b are area and centroidal distances of M/EI diagrams. It's used to analyze continuous beams by writing equations for each interior support and solving simultaneously. Settlement terms can be added for support movements.