330 THE ROYAL SOCIETY OF CANADA 



as / approaches plus or minus infinity, according as a is negative or 

 positive respectively. Such exponents as c are called by Poincaré the 

 "characteristic exponents." Eight of them arise in the problem under 

 consideration and thesp occur in pairs differing only in sign, this being 

 the case, as Poincaré has shown, ^ in all problems in mechanics in 

 which there is a conservative system. Two of the exponents are zero, 

 two are purely imaginary, and the remaining four are complex, having 

 their real parts different from zero. The solutions which approach 

 the equilibrium points as / approaches plus infinity, desingated in the 

 paper as "future orbits," are expanded ag power series in /^' and Z^' 

 where c,= — |a,l + \/— 1 &«, (* = 1, 2), and those which approach the 

 equilibrium points as t approaches minus infinity, designated "past 

 orbits," are expanded as power series in e"^^^ and e""^-'. Now to each 

 exponent such as Ci or C2 there corresponds one degree of freedom in 

 the solutions. Hence both the "future" and "past" orbits have two 

 degrees of freedom. 



The following geometric property of the orbits has been found. 

 Let the vertices of the equilateral triangle points be denoted by I and 

 II. Then there are two orbits approaching each vertex, the "future" 

 and the "past." A designation, such as "past II," means the orbit 

 which approaches the vertex II as / approaches minus infinity, and 

 similarly for the other orbits. Now it has been found that the orbit 

 "past II" may be obtained from "future I " by changing the sign of / 

 in the latter and then reflecting in the x-axis (rotating). The same 

 relation exists between "past I" and "future II." This is a sort of 

 converse of "history repeating itself" between the North and South. 



The paper concludes with numerical examples in which particular 

 values are assigned to the three bodies and the degrees of freedom 

 specified to be arbitrary displacements, parallel to the x-axis, of the 

 planetoid and Jupiter from the vertices of the triangle. The magni- 

 tudes of these displacements are restricted, however, by certain 

 convergence conditions. The orbits which have been computed are 

 drawn to scale. 



'Loc. cit., Vol. 1, Chap. IV. 



