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Astronomy. — "On periodic orbits of the type Hestia. By Dr. W. 

 de Sitter. (Communicated by Prof. J. C. K.ypteyni. 



The problem, of which some particular solutions will be treated 

 here, is the following. Two material points S and ./, having the 

 masses 1 and /«, move with uniform angular velocity n' = 1 in 

 circles in one and the same plane round their centre of gravity. 

 The constant distance SJ is adopted as unit of length. Another 

 material point 1\ with an infinitely small mass, moves in the same 

 plane under the influence of the Newtonian attractions of S and ./. 

 This is the problem which lias (for fi = 0.1) been so exhaustively 

 treated by Darwin in Vol. XXI of the Ada Mathematica. The parti- 

 cular solutions which are treated below are those in which the orbit 

 of' P is periodic and its limit for Dim. n = is an ellipse with a 

 small excentricity, described round S as a focus with a mean motion 

 not differing much from 3. If this limiting orbit (i.e. the undisturbed 

 orbit) is a circle, then the solution is, in Poincaré's phraseology, of the 



first sort (sorte), and its period is T= . If (he excentricity of the 



n — 1 



undisturbed orbit differs from zero, the solution is of the second 



sort, and the limiting value of the period for Lim. ;i = is Lim. 



T' = '2x. These solutions of the second sort are at the same lime 



of the second genus {(/rare) relatively to those of the first sort. 



The solutions of the first sort are the orbits of Darwin's "Planet 

 A". This family of orbits undergoes within the range here considered 

 a transition from stability to instability, which has been discussed 

 by 1'oincaré in an investigation contained in the articles 383 and 

 384 of his -Methodes Nouvelles" (Vol. Ill, p. 355—361). The 

 results there reached will be derived here by a different (and, as it 

 seems to me, simpler) reasoning. 



Darwin's work also presents an example of an orbit of the second 

 sort, viz. the orbit figured by him on page 281 and designated as 

 ■v = — .337. Although Poincahé proves the existence of solutions 

 of this kind, he seems to have overlooked the fact thai Darwin had 

 actually computed one of them. 



These solutions and their stability I wish to consider from the 

 point of view of the general theory developed by Poincakk in the 

 first and third volumes of the "Methodes Nouvelles". The following 

 is a summary of those general theorems, proved by Poincare, which 

 will be used here. They are true for every problem capable of 

 being reduced to two degrees of' freedom, containing one variable 

 parameter, and admitting for each value of this parameter a finite 



