On a Certain Integral of Problem of Ihree Bodies. 157 



be of great interest to see if these equations lead to expressions 

 for the orbits of heavenly bodies differing appreciably from 

 those hitherto in use. 



Summary. 



In this article it has been shown that the Einstein trans- 

 formation equations, and the other principles of non-Newtonian 

 mechanics, lead to a number of further transformation equa- 

 tions for acceleration, mass, rate of change of mass and force. 

 The transformation equations of force are identical with 

 those chosen by Planck. Two applications of the trans- 

 formation equations have been given. By combining them 

 with Coulomb's law, the expected equations have been derived 

 for the force with which an electric charge in uniform motion 

 acts on any other charge, and by combining them with 

 Newton's law a new expression has been derived for the 

 gravitational force with which a particle in uniform motion 

 acts on another particle. 



July 19, 1912. 



XVII. On a Certain Integral of the Problem of Three Bodies. 

 By Philip H. Ling, M.Sc* 



Introduction. 



IT is well known that the motion of three particles under 

 their mutual gravitational attractions cannot be com- 

 pletely determined. Certain combinations of the differential 

 equations can be integrated, but their number is not sufficient 

 to solve the problem, and it has been proved by Brims and 

 Painleve that no other integrals can exist. 



In astronomical problems it is usual to reduce the system 

 of three bodies to the simplest possible form. One body, P, 

 is supposed to have an infinitesimal mass, while the other 

 two, S and J, describe circles about each other. We have 

 then to determine the motion of P relative to the line SJ 

 and in the plane of motion of the two finite masses. This is 

 usually described as " the restricted problem of three bodies." 

 Only one integral can be obtained easily, and it has been 

 shown by Poincare f that no others exist which are one-valued 

 and regular. 



Now, it has recently occurred to the writer to attempt to 



* Communicated by the Author. The first part of this paper formed 

 a portion of a thesis accepted by the University of Bristol for the degree 

 of M.Sc. 



t An excellent account of the theorems of Bruns and Poincare will he 

 found in Prof. Whittaker's book on Analytical Dynamics. 



