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SCIENCE 



[N. S. Vol. XXIX. No. 733 



lier's paper no account was taken of 

 the imaginary centers of libration; the 

 analytical treatment was completed in this 

 particular by a note which showed that 

 there are imaginary centers about which 

 real orbits exist. Plummer has extended 

 Charlier's analysis to arbitrary fields of 

 force, and to terms of the second and third 

 orders in the developments. Schlitt has 

 reckoned five orbits to whose construction 

 Darwin referred as not belonging to the 

 category of simply periodic orbits, and for 

 that reason disregarded by him. With 

 Darwin's orbits Moulton has compared cer- 

 tain of his own, established by Poincare's 

 method of analytical continuation, and 

 arranged in power series rather than 

 Fourier series. Finally to Darwin's orbits 

 Stromgren has applied his conditions for 

 cusps and loops in the restricted three- 

 body problem; Stromgren has shown that 

 these singularities may be encountered in 

 every point in the plane, in the absolute 

 motion as well as in that referred to mov- 

 ing axes. 



Periodic orbits have been variously 

 classified. If the curves are reentrant 

 after a single period Darwin calls the 

 orbits "simply periodic"; all the orbits 

 considered by him have this property. 

 Hill has grouped them broadly into two 

 classes: the first contains those in which a 

 rotation of the whole system has taken 

 place; the second, those in which no such 

 rotation has occurred, but the longitudes of 

 the bodies and their distances have re- 

 turned to the same values. Poincare has 

 classified them elaborately into species, 

 classes and kinds, but as Charlier has 

 pointed out this classification is not ex- 

 haustive. The great majority of the orbits 

 referred to here belong to the first two 

 kinds, as distinguished by Poincare, that is, 

 they either have inclination and eccen- 

 tricity zero or inclination zero and eccen- 

 tricity not zero. Von Zeipel has published 



a thorough study of the solutions of the 

 third kind — that is, those having both in- 

 clination and eccentricity different from 

 zero — in which they are grouped in no 

 fewer than ten types and their stability 

 discussed by the aid of their characteristic 

 exponents. Whittaker has designed a 

 criterion for the discovery of periodic 

 orbits analogous to those theorems which 

 indicate the positions of the roots of an 

 algebraic equation. 



A matter of vital theoretical and prac- 

 tical import in the domain of periodic solu- 

 tions is the question of their stability. Fol- 

 lowing Poincare's lead, Brown has formu- 

 lated the sufficient conditions for stability 

 in the w-body problem as follows : first, that 

 the bodies never become infinitely distant 

 from one another ; second, that their mutual 

 distances never descend below a certain 

 limit; third, that each body passes an in- 

 finite number of times as near as we wish 

 to any point through which it has once 

 passed; fourth, that a small external dis- 

 turbance shall not affect the fulfillment of 

 these conditions. Poincare stated the first 

 three and investigated the third in detail; 

 numerical limits for the first and second 

 have been found by Haffel in a particular 

 case of the sun-earth-moon system. Levi- 

 Civita has worked out criteria in which the 

 stability is made to depend upon that of a 

 certain point transformation associated 

 with the periodic solution; these criteria 

 show the instability of certain orbits which 

 in the first approximation appear to be 

 stable; they indicate further that contrary 

 to accepted opinion a purely imaginary 

 characteristic exponent a does not always 

 single out a stable solution— the solution is 

 unstable if a/V — 1 is not commensurable 

 with the mean motion 2ir/r. Applying his 

 method to the restricted problem Levi- 

 Civita has found that solutions differing 

 little from circles and having a mean mo- 

 tion 1 4- 3//i are certainly unstable, thus 



