NOVELTY - Dispersion liquid (A) is dispersed in lithium transition metal complex oxide fine particle mixed in 0.1wt.% of hexa sodium metaphosphate aqueous solution so that 60-90% of laser light permeability is obtained. The particle ratio of dispersion liquid (B) in above composition having particle size less than 1microns, is 0.5% at the time of ultrasonic application for 5 minutes at frequency of 22.5kHz and output of 30W.    USE - Lithium transition metal complex oxide fine particle for positive electrode of lithium secondary battery used in motor vehicles, portable electronic device and communication apparatus.    ADVANTAGE - Due to low particle ratio, aggregation of the positive electrode materials is prevented and fine battery characteristic and favorable fine particle characteristic are obtained.    DETAILED DESCRIPTION - The diameter of secondary particle is 15microns and or less. The particle ratio of particle having size less then 1microns in the dispersion liquid (A) is 3% or less. The particle size distribution curve of the dispersion liquid (A) does not have point of inflexion in the range of 10-100 microns. The fine particle packing density and specific surface area of the lithium transition metal complex oxide which satisfies preset formula, is more than 1.6 g/cm3 abd 1.4 m2/g respectively. INDEPENDENT CLAIMS are also included for the following: v    (1) precursor for lithium transition metal complex oxide for lithium secondary battery positive electrode active materials, has atomic ratio of lithium with respect to transition element in the range of 0.001-0.5. The transition metal compound is nickel compound containing 5-50 wt% of nickel hydroxide. The median size of the primary particle is 0.25 microns or less. The particle ratio of the particle (C) of size 1mu or less at the time of ultrasonic application for 30 seconds is 0-25%.    (2) manufacturing method of precursor involves spray drying of wet grounded slurry containing lithium compound and transition metal compound and then baking obtained precursor;    (3) manufacturing method of lithium transition metal complex oxide fine particle;    (4) lithium secondary battery positive electrode, has active layer, lithium transition metal complex oxide fine particles and binder on collector; and    (5) lithium secondary battery has positive electrode, negative electrode which occlude lithium and electrolyte which contains lithium salt.