Abstract: This note proues the following result. Let a, b, y_a, y_b, y_a', y_b'be real numbers satisfying aab;y_a'≥0,y_b'≥0;y_a'+y_b'≤4((y_b-y_a/(b-a))) then the mo-notone increasing quadratic spline S(x)∈C¹[a,b]with knots at a,(a+b)/(2), b existsand is unique, and if y_a'+y_b'4((y_b-y_a)/(b-a)), such quadratic interpolating spline do-es not exist.According to this result the paper studies the existence of monotone cuadratic interpolating splines with more knots, and gives a computational method of smoothly Linking up the energy and pressure surfaces between different regions of state equations.