NOVELTY - A nickel cobalt composite hydroxide contains metal chosen from aluminum, magnesium, manganese, titanium, iron, copper, zinc and gallium, and has particle size distribution (D50) (D10) (D90). The difference of particle size distribution (D50) and particle size distribution (D10) with respect to particle size distribution (D50) is 0.30 or more. The difference of particle size distribution (D90) and particle size distribution (D50) with respect to particle size distribution (D50) is 0.3 or less.    USE - Nickel cobalt composite hydroxide is used for positive electrode active material for nonaqueous electrolyte secondary battery (all claimed) e.g. lithium-ion secondary battery. Uses include but are not limited to mobile telephone, notebook personal computer, hybrid vehicle, electric vehicle, gasoline engine, and diesel engine.    ADVANTAGE - The nickel cobalt composite hydroxide has excellent particle size distribution, and provides nonaqueous electrolyte secondary battery having excellent cycle characteristics, and is economically manufactured with high productivity by effective usage of raw material.    DETAILED DESCRIPTION - A nickel cobalt composite hydroxide has particle size distribution (D10, D50, D90), and is of formula: Ni1xyCoxMy(OH)2 (I). The difference of particle size distribution (D50) and particle size distribution (D10) with respect to particle size distribution (D50) is 0.30 or more. The difference of particle size distribution (D90) and particle size distribution (D50) with respect to particle size distribution (D50) is 0.3 or less.    M=aluminum, magnesium, manganese, titanium, iron, copper, zinc or gallium;    x=0.05-0.5;    y=0-0.1;and    x+y=0.05-0.5.    INDEPENDENT CLAIMS are included for the following:    (1) manufacture of nickel cobalt composite hydroxide, which involves supplying aqueous solution containing nickel and cobalt, aqueous solution containing ammonium-ion supply material and caustic-alkali aqueous solution, respectively, to reaction tank, reacting, continuously extracting slurry containing nickel cobalt composite hydroxide, separating extracted nickel cobalt composite hydroxide to large particle size material and small particle size material, and continuously recirculating small particle size material to reaction tank;    (2) apparatus for manufacturing nickel cobalt composite hydroxide, which has reaction tank (1), unit (3) for supplying aqueous solution, unit (4) for supplying aqueous solution containing ammonium-ion, unit (5) for supplying caustic-alkali aqueous solution, unit for continuously extracting nickel cobalt composite hydroxide, separation apparatus (7) for separating extracted nickel cobalt composite hydroxide, and recirculation apparatus (9) for continuously recirculating separated small particle size material to the reaction tank;    (3) positive electrode active material, which has a layer structure comprising hexagonal-type lithium-nickel cobalt complex oxide of formula: Li1+uNi1xyCoxMyO2 (II). The nickel cobalt composite oxide has particle size distribution (D50, D10, D90). The difference of particle size distribution (D50) and particle size distribution (D10) with respect to particle size distribution (D50) is 0.30 or more. The difference of particle size distribution (D90) and particle size distribution (D50) with respect to particle size distribution (D50) is 0.3 or less;    (4) manufacture of positive electrode, which involves baking nickel cobalt composite hydroxide or nickel cobalt composite oxide at 300-700 degrees C in oxidative atmosphere, mixing with lithium compound, obtaining lithium mixture, and baking the obtained lithium mixture at 600-850 degrees C in oxidative atmosphere; and    (5) non-aqueous electrolyte battery, which has a positive electrode containing the positive electrode active material as positive electrode material, a negative electrode, a separator, and a nonaqueous electrolyte.    u=0-0.3.    DESCRIPTION OF DRAWING(S) - The drawing shows a schematic view of apparatus for manufacturing nickel cobalt composite hydroxide.    Reaction tank (1)    Unit for supplying aqueous solution (3)    Unit for supplying ammonium ion solution (4)    Unit for supplying caustic-alkali aqueous solution (5)    Separation apparatus (7)    Recirculation apparatus (9)