NOVELTY - The pH of aqueous solution (i) containing metallic compound comprising nickel, metallic compound containing manganese and an ammonium-ion supply material is controlled to 12-13.4 measured at liquid temperature of 25 degrees C, and the aqueous solution is nucleated. The pH of aqueous solution containing nucleus is controlled to 10.5-12 measured at liquid temperature of 25 degrees C, and aqueous solution containing nucleus is subjected to particle growth process, to obtain nickel-manganese composite hydroxide particles (I).    USE - Manufacture of nickel-manganese composite hydroxide particles used for manufacturing positive-electrode active material for nonaqueous electrolyte secondary battery (all claimed) for notebook personal computer, electric vehicle, hybrid vehicle and mobile telephone.    ADVANTAGE - The nonaqueous electrolyte secondary battery containing the nickel-manganese composite hydroxide particles has excellent durability, thermal stability and charging/discharging characteristics, and high capacity.    DETAILED DESCRIPTION - The pH of aqueous solution (i) containing metallic compound comprising nickel, metallic compound containing manganese and an ammonium-ion supply material is controlled to 12-13.4 measured at liquid temperature of 25 degrees C, and the aqueous solution is nucleated. The pH of aqueous solution containing nucleus is controlled to 10.5-12 measured at liquid temperature of 25 degrees C, and aqueous solution containing nucleus is subjected to particle growth process, to obtain nickel-manganese composite hydroxide particles of formula Ni1-x-yMnxMy(OH)a (I).    M = titanium, vanadium, chromium, zirconium, niobium, molybdenum, or tungsten;    x = 0.45-0.55;    y = 0-0.05;    a = 2+b; and    b = 0-0.5.    INDEPENDENT CLAIMS are included for the following:    (1) manufacture of lithium-nickel-manganese complex oxide aode active material of formula Lit(Ni1-x-yMnxMy)2-tO2 or non-aqueous electrolyte secondary batteries    positive electrode (3a), which involves heat-processing nickel-manganese composite hydroxide particles at 500-700 degrees C, then mixing with a lithium compound and baking at 800-950 degrees C;    (2) positive-electrode active material; and    (3) nonaqueous electrolyte secondary battery.    M = titanium, vanadium, chromium, zirconium, niobium, molybdenum, or tungsten;    x = 0.45-0.55;    y = 0-0.05; and    t = 1.00-1.20.    DESCRIPTION OF DRAWING(S) - The drawing shows the schematic view of the coin-type battery. (Drawing includes non-English language text)    Coin-type battery (1)    Case (2)    Electrode (3)    Positive electrode (3a)    Separator (3c)