This is an annotated translation from Latin of E327 -- De motu rectilineo trium corporum se mutuo attrahentium (“On the rectilinear motion of three bodies mutually attracting each other”). In this publication, Euler considers three bodies lying on a straight line, which are attracted to each other by central forces inversely proportional to the square of their separation distance (inverse-square law). Here Euler finds that the parameter that controls the relative distances among the bodies is given by a quintic function.
Bistafa, Sylvio R.
"On the Rectilinear Motion of Three Bodies Mutually Attracting Each Other,"
Euleriana: 1(2), p. 188, Article 7.
Available at: https://scholarlycommons.pacific.edu/euleriana/vol1/iss2/7