NOVELTY - Lithium-iron phosphate or lithium-iron silicate is obtained by hydrothermally reacting a mixed slurry containing transition metal compound, silicic acid compound, lithium compound, and water, in a pressure-resistant container (2). A synthesis container (1) containing the mixed slurry is installed in the pressure-resistant container. The mixed slurry is extracted with a pump (4) out of the pressure-resistant container from a piping (3) linked to the synthesis container and reacted by stirring after returning by a different piping (5).    USE - Manufacture of transition-metal-containing olivine type silicate compound other than lithium-iron silicate, for use as positive-electrode material in lithium ion secondary batteries (claimed) used in portable electronic device, hybrid vehicles, and electric vehicles.    ADVANTAGE - The method enables manufacture of lithium-iron phosphate or lithium-iron silicate having uniform composition, fine and uniform particle size, and high purity, with high yield. The obtained compound has high capacity and excellent charging/discharging characteristics.    DETAILED DESCRIPTION - Lithium-iron phosphate or lithium-iron silicate is obtained by hydrothermally reacting a mixed slurry containing compound of transition metal (M) (iron, nickel, cobalt, aluminum, zinc, vanadium, zirconium or manganese), silicic acid compound, lithium compound, and water, in a pressure-resistant container. The synthesis container containing the mixed slurry is installed in a pressure-resistant container. The mixed slurry is extracted with a pump out of the pressure-resistant container from a piping linked to the synthesis container. Hydrothermal reaction is carried out in the synthesis container by stirring after returning the mixed slurry discharged from the pump by a piping different from the piping linked to the synthesis container.    DESCRIPTION OF DRAWING(S) - The drawing shows a schematic view of the reaction apparatus used in the manufacture of transition-metal-containing olivine type silicate compound. (Drawing includes non-English language text)    Synthesis container (1)    Pressure-resistant container (2)    Piping (3,5)    Pump (4)    Heating steam (6)