NOVELTY - An aqueous solution containing metallic compound containing transition metal and/or ammonium ion supply component is subjected to nucleation by controlling pH value to 12-14 at 25 degrees C of liquid temperature standard, to form nuclei. The resultant product is subjected to particle growth by controlling pH value of aqueous solution containing the nuclei to 10.5-12 at 25 degrees C of liquid temperature standard and nuclei are grown, to obtain transition metal composite hydroxide particles. In the particle growth process, non-oxidizing atmosphere is switched to oxidizing atmosphere.    USE - Manufacture of transition-metal composite hydroxide particles used for forming positive electrode active material for nonaqueous electrolyte secondary battery (all claimed). Uses include but are not limited to hybrid electric vehicle, plug-in hybrid electric vehicle, battery-type electric vehicle, mobile telephone and notebook computer.    ADVANTAGE - The method enables easy manufacture of transition metal composite hydroxide particles. The nonaqueous electrolyte secondary battery obtained using the positive electrode active material containing transition metal composite hydroxide particles has excellent capacitance characteristics, output characteristics and cycle characteristics.    DETAILED DESCRIPTION - An aqueous solution containing metallic compound containing transition metal and/or ammonium ion supply component is subjected to nucleation by controlling pH value to 12-14 at 25 degrees C of liquid temperature standard, to form nuclei. The resultant product is subjected to particle growth by controlling pH value of aqueous solution containing the nuclei to 10.5-12, which is less than the pH value of nucleation at 25 degrees C of liquid temperature standard and nuclei are grown, to obtain transition metal composite hydroxide particles. The initial stages of the nucleation and the particle growth are performed in non-oxidizing atmosphere, with oxygen concentration of 5 vol.% or less. In the particle growth process, non-oxidizing atmosphere is switched to oxidizing atmosphere when the oxygen concentration is more than 5 vol.%, and switched again to the non-oxidizing atmosphere at least once. The initial stage particle growth process time in non-oxidative atmosphere is 5-35% with respect to total particle growth process time. The atmosphere control is switched from oxidative atmosphere to non-oxidizing atmosphere once when oxygen concentration is 5 vol.% or less. The crystallization reaction time in the oxidative atmosphere is 3-20% with respect to total particle growth process time. INDEPENDENT CLAIMS are included for the following:    (1) transition metal composite hydroxide particles, which comprises secondary particles formed by aggregating plate-shaped primary particles, and fine primary particles smaller than plate-shaped primary particles. The secondary particles have center portion formed by aggregating the plate-shaped primary particles. The transition metal composite hydroxide particles have a laminated structure obtained by laminating low density portion formed in the outer side of center portion by aggregating fine primary particles and high-density portion formed by aggregating the plate-shaped primary particles. The average value of ratio of the outer diameter of center portion to the particle size of the secondary particles is 30-80%, and the average value of ratio of the thickness of the radial direction of the high-density portion to the particle size of secondary particles is 5-25%. The average particle diameter of secondary particles is 1-15 mu m. The parameter obtained by calculating the ratio of breadth of particle size distribution (d90-d10) and average particle diameter of the secondary particles is 0.65 or less;    (2) manufacture of positive electrode active material, which involves mixing transition metal composite hydroxide particles and lithium compound, forming lithium mixture and baking the lithium mixture at 650-980 degrees C in an oxidative atmosphere;    (3) positive electrode active material, which comprises lithium-transition metal composite oxide particles having crystal structure of hexagonal system having layer structure, and secondary particles formed by aggregating primary particles. The secondary particles have center portion of solid structure or hollow structure, and provides the space portion which does not contain primary particles, outer shell portion electrically connected to the center portion, and inner shell portion between space portion and outer shell portion. The average value of ratio of the outer diameter of center portion to the particle size of the secondary particles is 30-80%, and the average value of ratio of the thickness of the radial direction of the outer shell portion to the particle size of secondary particles is 5-25%. The average particle diameter of secondary particles is 1-15 mu m. The parameter obtained by calculating the ratio of breadth of particle size distribution (d90-d10) and average particle diameter of the secondary particles is 0.7 or less; and    (4) nonaqueous electrolyte secondary battery, which consists of positive electrode containing positive electrode active material, negative electrode, separator and nonaqueous electrolyte.