NOVELTY - Nucleation is carried out while controlling an aqueous solution for nucleation containing an ammonium-ion supply component and a metal compound containing nickel to have a pH of 12-13.4 at 25 degrees C. The nucleus is grown while controlling an aqueous solution for particle growth containing the nucleus obtained by nucleation process to have a pH of 10.5-12 (less than pH in nucleation process) at 25 degrees C, to obtain particles of nickel composite hydroxide.    USE - Manufacture of nickel composite hydroxide particles used in manufacture of positive-electrode active materials of nonaqueous electrolyte secondary batteries (all claimed) e.g. lithium ion secondary batteries. Uses include but are not limited to mobile telephones, notebook type personal computers, electric vehicles, and hybrid vehicles which use gasoline engines and diesel engines.    ADVANTAGE - The method enables manufacture of nickel composite hydroxide particles having small and uniform particle diameter. Since the nucleus is grown homogeneously, the nickel composite hydroxide particles have nanoparticle size distribution. Since the aqueous solution for particle growth is obtained by adjusting the pH of aqueous solution used for nucleation, after completion of nucleation, transfer to particle growth process can be carried out rapidly. The particles can be grown with high concentration. The batteries obtained using positive electrodes manufactured using the nickel composite hydroxide particles have low electrode resistance and suppressed electrode deterioration even during repeated charging/discharging. The compact high-power batteries also have excellent safety, durability, and output characteristics. The positive electrodes have reduced reaction resistance and reduction in initial-stage discharge capacity.    DETAILED DESCRIPTION - Nucleation is carried out while controlling an aqueous solution for nucleation containing an ammonium-ion supply component and a metal compound containing nickel to have a pH of 12-13.4 at 25 degrees C. The nucleus is grown while controlling an aqueous solution for particle growth containing the nucleus obtained by nucleation process to have a pH of 10.5-12 (less than pH in nucleation process) at 25 degrees C, to obtain particles of nickel composite hydroxide of formula: Ni1-x-yCoxMy(OH)2+a.    x=0-0.2;    y=0-0.15;    x+y=less than 0.3;    a=0-0.5;and    M=Mg, Al, Ca, Ti, V, Cr, Mn, Zr, Nb, Mo and/or W.    INDEPENDENT CLAIMS are included for the following:    (1) nickel composite hydroxide particles, which are spherical secondary particles formed by aggregating plate-shaped primary particles containing nickel composite hydroxide. The secondary particles have an average particle diameter of 3-7 mu m. The parameter ((d90-d10)/average particle diameter) which shows the spread of particle size distribution is 0.55 or less;    (2) manufacture of positive-electrode (3a) active material containing a lithium-nickel-cobalt complex oxide of formula: LitNi1-x-yCoxMyO2, which involves heat-processing the nickel composite hydroxide particles, mixing the heat-processed particles with a lithium compound to form a mixture, and baking the mixture at 700-850 degrees C; and    (3) positive-electrode active material having ((d90-d10)/average particle diameter) value of 0.6 or less, and an average particle diameter of 2-8 mu m.    t=0.95-1.15.    DESCRIPTION OF DRAWING(S) - The drawing shows a schematic sectional view of a coin-type battery. (Drawing includes non-English language text)    Coin-type battery (1)    Case (2)    Electrode (3)    Positive electrode (3a)    Negative electrode (3b)