OnLagrange's triangular solution of the problem of three bodies

Conclusion

The equality of the mutual distances of all the particles is of basic importance in the expansion ofLagrange's theorem to general central forces. It seems to me that in more general cases the corresponding generalizations of propositions 2.4. and 2.5. do not hold. I suppose that, there is a close relation, perhaps a kind of conditional equivalence, between the concepts of homographic solution, perpetual central configuration and homogeneous force, but I have not been able to prove my supposition, except under additional assumptions which decrease the interest in the statements. I hope somebody else will be more successful.