NOVELTY - A raw material aqueous solution containing nickel and manganese, aqueous solution containing ammonium-ion donor, and alkaline solution are supplied to a reaction tank and mixed to obtain aqueous reaction solution and nickel manganese composite hydroxide particle is crystallized. The oxygen concentration in the reaction tank is controlled to 3 vol.% or less. Temperature of the aqueous reaction solution is controlled to 35-60 degrees C, and nickel ion concentration is controlled to 1000 mg/L or more, to obtain nickel manganese composite hydroxide particle (1).    USE - Manufacture of nickel manganese composite hydroxide particle used for manufacture of positive electrode active material for positive electrode of non-aqueous electrolyte secondary battery (all claimed) for portable electronic device e.g. notebook-type personal computer and mobile telephone terminal, hybrid vehicles and electric vehicles.    ADVANTAGE - The method enables efficient manufacture of nickel manganese composite hydroxide particle with improved circularity. The composite hydroxide particles provide positive electrode active material with improved fillability. A non-aqueous electrolyte secondary battery has excellent battery capacity and cycle characteristics by using the positive electrode active material.    DETAILED DESCRIPTION - A raw material aqueous solution containing nickel and manganese, aqueous solution containing ammonium-ion donor, and alkaline solution are supplied to a reaction tank and mixed to obtain aqueous reaction solution and nickel manganese composite hydroxide particle is crystallized. The oxygen concentration in the reaction tank is controlled to 3 vol.% or less. Temperature of the aqueous reaction solution is controlled to 35-60 degrees C, and nickel ion concentration is controlled to 1000 mg/L or more, to obtain nickel manganese composite hydroxide particle of formula (1): NixMnyMt(OH)2+a. Nickel manganese composite hydroxide particles contain secondary particles formed by aggregating several primary particles.    x=0.05-0.95;    y=0.05-0.95;    t=0-0.2;    x+y+t=1;    a=0-0.5;and    M=Co, Ti, V, Cr, Zr, Nb, Mo, Hf, Ta and/or W.    INDEPENDENT CLAIMS are included for the following:    (1) manufacture of positive electrode active material, which involves mixing lithium compound and heat processed particles obtained by heat-processing the nickel manganese composite hydroxide particles to obtain a lithium mixture, and baking the lithium mixture at 720-1000 degrees C in an oxidative atmosphere;    (2) positive electrode active material of formula: Li1+uNixMnyMtO2, where u is -0.05 to 0.50, which consists of hexagonal type lithium nickel manganese complex oxide particles which have layered structure, and which have average circularity of 70% or more of particle size of average particle diameter of 0.82 or more, and tap density of 2.20 g/cm3 or more; and    (3) non-aqueous electrolyte secondary battery, which has a positive electrode containing the positive electrode active material, negative electrode, separator and non-aqueous electrolyte.