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number of periodic solutions. It need hardly lie mentioned that 

 their valency is restricted to a certain domain of the several variable 

 quantities of the problem, of which it will however not be necessary 

 to transgress the limits. 



A periodic solution is completely determined by the values of the 

 parameter and of one constant of integration or "element". The 

 periodic solutions occur in families, the members of which are classi- 

 fied according to increasing or decreasing values of the parameter. 

 These families may be graphically represented by curves (x , /? ) = 0, 

 where x is the parameter of the problem and ft the determining 

 element. 



The stability or instability is determined by a certain quantity ft, 



which is by Poincaré called the characteristic exponent. If the 



2 k i 

 period is T, then values of a differing' by a multiple of — — , must 



be considered as identical. The following three cases are possible: 



ft 7' purely imaginary the solution is sta ble 



uT real the solution is even/;/ unstable 



aT complex, with imaginary part =ni: the solution is unevenly unstable 1 ). 



A solution having the period T can as well be conceived to have 

 the period T' = 2 T. If it is unevenly unstable with reference to 

 the period T, it is evenly unstable with reference to the period T' . 



Within each family the exponent « and the period T vary conti- 

 nuously with the parameter x. The product aT and the differential 



(10 



coefficient become equal to zero for the same values ot x. The 



dp 



curve <p = then either has a multiple point, or is tangent to a 



line x = const. The family splits into two branches, or, which 



comes to the same thing, two families have one member in common. 



If (*o &>) i g *' ie point representing this common member, then we 



have the following rules. 



The number of branches of the curve = (i.e. the number of 

 families of periodic, solutions) for x^> x differs by an even number 

 from the number of branches for x <^ x„. 



The branches which part from the point (x /?„) towards the direction 

 of increasing x are alternately stable and evenly unstable ''). The 



i) The names even and uneven instability have been introduced by Darwin. 

 Poincaré distinguishes them as instability of the first and second "classe". The 

 relation of Darwin's quantity c to the exponent j. is given by the formula nT = inc. 



3 ) To avoid circumlocution I speak of "stable and unstable branches", meaning 

 branches whose points represent stable and unstable solutions respectively. 



