1064 CHARLIER, ON PERIODIC ORBITS. 



hence 



?1 = ^2 = 1 • 



This point (a, b) lies on the vertex of the equilateral 

 triangle described om m.^m.^. 



There are evidently two such points, and tliese I "will call 

 (a^, b^) and (a-, />.), and we have 



?', — ^2 1 f.1 



n — ri — _! ^ — 1 . L 



(9) ^ ^ ^ 2 ^1 + ^. 



The points corresponding to the Solution (8) must have 

 5 = 0, and are thence situated on the x axis. The value of a 

 for these points are deterinined by the equation 



(10) dQ.a-r,^d^a_^^^^^ 



In calculating the roots of this equation we may properly 

 distinguish between three different cases: 



1) « < — r^ 



2) — ^2 < « < r^ 



3) 7\ < a . 

 And it is now: 



1) a — ?'i = — Oj ; « + ^2 = — ?2 



2) a — rj = — (», ; a + ^'o = + Q^ 



3) rt — *'i = + ?i 5 a + n = + ^2 • 



The equation (10) assumes in these 3 cases the following 

 form : 



9X ^^ ^-^ A. 1 



^) ä^ + ^^ = ^' 5 Ci = ?" — 1 • 



