In this report, following problem of steady conjugate heat transfer with conduction and free convection is treated by analytical method, to which the local similarity solution is applied. The left side surface of vertical flat plate with finite thickness is maintained at uniform temperature or uniform heat flux, and the right side surface is in contact with fluid in which laminar free convection heat transfer takes place. The differences of thermal condition between each surface are made clear by the use of the nondimensional factor k_f/k_s・d/l for uniform temperature at the left side surface and k_s/k_f・d/l for uniform heat flux.