NOVELTY - Manufacture of nickel-cobalt-composite hydroxide particles involves mixing metal salt aqueous solution containing cobalt and aqueous solution containing ammonium-ion supply component, and controlling the pH of the obtained mixture, nucleating the mixture in oxidative atmosphere, controlling the pH value of the aqueous solution containing nucleus, supplying metal salt aqueous solution and aqueous solution and carrying out particle growth process in oxygen concentration of 0.5 vol.% or less for nucleus growth.    USE - Manufacture of nickel-cobalt-composite hydroxide particles used for manufacturing positive electrode active material for nonaqueous electrolyte secondary battery (all claimed). Uses include but are not limited to notebook-type personal computer, mobile telephone terminal, hybrid electric vehicle, plug-in hybrid electric vehicle, and battery-type electric vehicle.    ADVANTAGE - The method enables easy and economical manufacture of nickel-cobalt-composite hydroxide particles and provides safe and light-weight nonaqueous electrolyte secondary battery having excellent capacitance and charging/discharging characteristics.    DETAILED DESCRIPTION - Manufacture of nickel-cobalt-composite hydroxide particles involves mixing metal salt aqueous solution containing cobalt and aqueous solution containing ammonium-ion supply component, and controlling the pH of the obtained mixture to 12-14 at liquid temperature of 25 degrees C, nucleating the mixture in oxidative atmosphere with the oxygen concentration more than 10 vol.%, controlling the pH value of the aqueous solution containing nucleus generated at the nucleation process to 10.5-12 at liquid temperature of 25 degrees C, supplying metal salt aqueous solution containing nickel and cobalt, and aqueous solution containing ammonium ion supply component, and carrying out particle growth process in oxygen concentration of 0.5 vol.% or less for nucleus growth, and in which the cobalt ion content rate in the metal ion of metal salt aqueous solution is 70 mol% or more, and controlling cobalt ion content rate to 4-11 mol% with respect to the total amount of the metallic element contained in the metal salt aqueous solution for particle growth which supplies the production amount of the nucleus in the nucleation process through the particle growth process. The nickel cobalt composite hydroxide particles are of formula: NixCoyMnzMt(OH)2+a (I).    x+y+z+t=1;    x=0.30-0.90;    y=0.10-0.40;    z=0-0.40;    t=0-0.10;and    M=additional elements chosen from Mg, Ca, Al, Ti, V, Cr, Zr, Nb, Mo and W.    INDEPENDENT CLAIMS are included for the following:    (1) nickel-cobalt composite hydroxide particles, which comprises plate-shaped primary particles formed by aggregating spherical-shaped secondary particles having average particle diameter of 1-5 mu m, and center portion including high cobalt-containing portion, and has particle size distribution index of 0.65 or less, and porosity of 15% or less;    (2) manufacture of positive electrode active material, which involves mixing lithium compound with nickel cobalt composite hydroxide particle and forms lithium mixture and baking the lithium mixture at 800-1000 degrees C in oxidative atmosphere, and in which the positive electrode active material comprises lithium nickel cobalt complex oxide particles of formula: Li1+uNixCoyMnzMtO2 and has a layer structure; and    (3) nonaqueous electrolyte secondary battery, which comprises positive electrode containing the positive electrode active material, negative electrode, separator, and nonaqueous electrolyte.    u=0.05-0.50.