Jantjart 15, 1909] 



SCIENCE 



91 



treatment of the three-body problem. 

 Esclagnon and Bohl have indicated ap- 

 plications of quasi-periodic functions to the 

 ordinary problem; special cases in which 

 the masses vary with the time have been 

 considered by Mestehersky; Laves has 

 studied the integrals when the forces de- 

 pend upon the coordinates and their de- 

 rivatives of the first two orders ; and Ebert 

 has taken up the problem in space of any 

 number of dimensions. 



Bertrand inverted the problem of two 

 bodies by proposing to find the law of force 

 under which a body, whatever may be its 

 initial position and velocity, always de- 

 scribes a conic section. This' inverse prob- 

 lem was solved independently by Bertrand, 

 Darboux and Halphen; and extended by 

 Dainelli to general curve trajectories. 

 Stephanos has recently given another gen- 

 eralization of Bertrand 's problem by in- 

 eluding in the discussion the case in which 

 the force has not necessarily an unique 

 direction at every point of the conic sec- 

 tion. This problem in turn has been gen- 

 eralized to conditions) which include the 

 conic section trajectories as special cases. 

 GrifSn observed that the law of force under 

 which a given curve is described as a cen- 

 tral orbit can not be determined uniquely 

 if only the position of the center of force 

 be known. Oppenheim gave to Bertrand 's 

 problem a new treatment which included 

 the case of finding the central conservative 

 forces under which three bodies of arbi- 

 trary mass describe given plane curves. 



A further generalization of Bertrand 's 

 problem presents itself in the problem of 

 finding the forces of a central conservative 

 system capable of maintaining a system of 

 m particles on as many prescribed but arbi- 

 trary orbits in a space of n dimensions. 

 The resolution of this problem shows that 

 the central conservative character of the 

 motion and the equations of the orbits are 

 necessary and sufficient to determine the 



components of the velocities, only in the 

 case of in{n-\-l) bodies, and the com- 

 ponents of the forces only in that of 2m — 1 

 bodies. From this point of view the plane 

 three-body problem possesses an unique 

 generality of its own, in that it is the only 

 case in which all the elements of the me- 

 chanics of the problem are completely de- 

 terminate when the arbitrary plane curves 

 described by the bodies under central con- 

 servative forces are given. This circum- 

 stance has been turned to account in the 

 construction of new integrable problems of 

 three bodies under laws of force involving 

 only the masses and the mutual distances 

 of the bodies. 



Edgab Odell Lovett 

 The Rice Institute, 

 Houston, Texas 



THE PHYLETIC IDEA IN TAXONOMY ■" 

 To-DAT every botanist is an evolutionist. 

 It may well be that we have not yet agreed 

 as to the details— as to the particular man- 

 ner in which modifications were effected — 

 whether they were by slow and almost im- 

 perceptible deviations from the parental 

 type, or those more marked variations that 

 we are in the habit to-day of calling "mu- 

 tants." Some of us may lay more stress 

 upon the "survival of the fittest," others 

 upon the "survival of the unlike." For 

 some the "struggle for existence" may ac- 

 count for the diversity of plant forms, 

 while others see in "adaptation" the ex- 

 planation of the same diversity. To some 

 the "inherent tendency" in plants to vary 

 is a potent factor, while for others all vari- 

 ation is a result of "environment." Tet 

 with all this diversity of opinion as to 

 details there is a practical unanimity as to 

 the acceptance of the general doctrine of 

 evolution. It may be asserted without fear 



'Address of the vice-president and chairman of 

 Section G — Botany — of the American Association 

 for the Advancement of Science, Baltimore, 1908. 



