# Where can I find Lagrange’s original equilateral triangular solution for arbitrary masses?

This answer to What kind of triangle is formed by three unequal masses in a circular restricted three body orbit? explains that

In the Newtonian limit, an equilateral 3-body solution exists for any combination of masses

and links to Triangular solution to general relativistic three-body problem for general masses which stars with "Lagrange’s equilateral triangular solution for the three-body problem" and looks at relativistic ("post-Newtonian") effects.

In the circular restricted three body problem of Lagrangian points fame one body is a "test mass" with no significant effect on the other two, but apparently three massive bodies can also have a circular orbit solution where they still form an equilateral triangle, as $$L_4$$ and $$L_5$$ do in the CR3BP.

Where did Lagrange first write about the equilateral triangular solution for the three-body problem where all three bodies have non-zero mass?

• This link may prove useful for you. The name of the book is "Three Body Dynamics and Its Applications to Exoplanets". Link to google books – Nachiket Kulkarni May 11 '20 at 15:10
• It seems your title question is not exactly the same as your body question. Or maybe I'm misinterpreting that final sentence as "when ...." ? – Carl Witthoft May 12 '20 at 12:55
• @CarlWitthoft I'm looking for the original writing, did Lagrange write it in a scientific paper, or in a letter, or in a journal? Thus "where" was it written. – uhoh May 12 '20 at 14:36
• Lagrange points /Wikipedia article ref.4 perhaps, it's a link to his Oeuvres t.6 p. 229 and it is printed with a note "Prix de l'Academie royale des sciences de paris tome IX 1772" which could be checked. – sand1 Sep 20 '20 at 8:44
• @sand1 Thank you! Yes "any three masses" and I should have caught that. I see a link here to page 229 in gallica and I did study French 45 years ago :-) I'll dig in and give it a try. It's possible that no English translation exists. – uhoh Sep 20 '20 at 9:42