On the Regularization of the Problem of Three Bodies

It is not my intention to enter here into a discussion of the general problem of three bodies, the first particular solutions of which were found by Lagrange before the year 1772, and were reproduced by Laplace in his Mechanique Celeste. To determine the mutual distances ot the bodies Lagrange found three equations, of which one was of the third order, and the other two were each of order two, thus making the whole problem of the seventh order. No new solutions were discovered after those of Lagrange, until Hill developed his Lunar Theory and found solutions of greater practical value than those of Lagrange himself. The most important advance was made by Poincare, who, assuming the masses of two of the bodies to be small compared to that of the third, proved that there were an infinite number of sets of initial conditions for which the motion was periodic. In 1899 Sir George Darwin assumed that one of the three masses was infinitesimal, and that the finite masses revolved in circles.