关键词: N-body problems, conjugate-nest consisted of one regular tetrahedron and one
regular octahedron, central configurations, existence and uniqueness

Abstract: A new case configuration in R³, the conjugate-nest consisted of one regular tetrahedron and one regular octahedron is discussed. If the configuration is a central configuration, then all masses of outside layer are equivalent, the masses of inside layer are also equivalent. At the same time the following relation between ρ(r = √3/3 ρ is the radius ratio of the sizes) and mass ratio τ = m~/m must be satisfied τ = ...  and for any mass ratio τ, when mass ratio τ is in the open interval (0,0.03871633950 ···), there exist three central configuration solutions(the initial configuration conditions who im- ply hamagraphic solutions) corresponding radius ratios are r₁,r₂, and r₃, two of them in the interval (2.639300779··· ,+∞) and one is in the interval (0.7379549890···, 1.490942703···). when mass ratio τ is in the open interval (130.8164950··· ,+∞), in the same way there have three corresponding radius ratios, two of them in the interval (0,0.4211584789···) and one is in the interval (0.7379549890···, 1.490942703···). When mass ratio τ is in the open interval (0.03871633950···, 130.8164950···), there has only one solution r in the interval (0.7379549890···, 1.490942703···). 

Key words: N-body problems, conjugate-nest consisted of one regular tetrahedron and one
regular octahedron, central configurations, existence and uniqueness