This development was given a more mathematical form by Einstein, who noted that the phase function for the waves, θ θ ( J , x ) , in a mechanical system should be identified with the solution to the Hamilton–Jacobi equation, an equation which William Rowan Hamilton believed to be a short-wavelength limit of a sort of wave mechanics in the 19th century. Schrödinger then found the proper wave equation which matched the Hamilton–Jacobi equation for the phase; this is now known as the Schrödinger equation.