On Dziobek Special Central Configurations

We study the special central configurations of the curved N-body problem in S^3. We show that there are special central configurations formed by N masses for any N >2. We then extend the concept of special central configurations to S^n, n>0, and study one interesting class of special central configurations in S^n, the Dziobek special central configurations. We obtain a criterion for them and reduce it to two sets of equations. Then we apply these equations to special central configurations of 3 bodies on S^1, 4 bodies on S^2, and 5 bodies in S^3.