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    I think it is saying that you have two particles on a sphere which, at distance 'd' (and only d) apart, have forces in equilibrium. At any other distance the forces are zero.

    1) Adding a third particle 'P' also at distance 'd' from the other two which are now all in equilibrium, then the original particles must be balanced about 'P'.

    2) Any number of particles for which the law holds true and considering the forces acting on a chosen particle P, is termed a 'configuration of particles balanced about P'.

    3) When any given configuration of particles are in equilibrium, then the whole 'configuration' is simply 'balanced'.

    So your original statement for the configuration where only one distance 'd' for those three particles on the equator can be in latitudinal balance and 'd' is therefore 2pi/3 rad. However they cannot be in balance in any other direction because it is not 2pi/3 in any other direction so the forces are zero.

    If you add a fourth particle on the equator, then d on the equator is also now violated (pi/2 rad) and the forces are zero.
 
 
 
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