Starting with the governing equations in terms of displacements of 3D elastic media,the solutions to displacement components and their first derivatives are obtained by the application of a double Fourier transform and an order reduction method based on the Cayley-Hamilton theorem.Combining the solutions and the constitutive equations which connect the displacements and stresses,the transfer matrix of a single soil layer is acquired.Then,the state space solution to multilayered elastic soils is further obtained by introducing the boundary conditions and continuity conditions between adjacent soil layers.The numerical analysis based on the present theory is carried out,and the vertical displacements of multilayered foundation with a weak and a hard underlying stratums are compared and discussed.