108 



SCIENCE. 



[N. S. Vol. XVII. No. 420. 



The method of Poineare is then used to de- 

 sign other periodic solutions; by an easy 

 reduction the equations become amenable 

 to the treatment proposed by Oppenheim in 

 the corresponding case of three bodies. 



The Problems of Three or More Bodies with 

 Prescribed Orbits: Professor E. 0. 

 LovETT, Princeton University. 

 This paper has points of contact with 

 and generalizes certain theorems due to 

 Bertrand, Darboux, Halphen and Oppen- 

 heim. Two problems are studied : 



1. The determination of the curves 

 which three bodies may describe under 

 central forces possessing a force function, 

 this function to have a form assigned in 

 advance. The results, other than those 

 which are well known, are transcendental. 



2. The determination of forces which 

 maintain the motions of any number of 

 bodies in prescribed orbits independent of 

 initial conditions in a space of any num- 

 ber of dimensions, the forces assumed cen- 

 tral. It appears that in general a certain 

 number of the forces may be chosen arbi- 

 trarily. In ordinary three-dimensional 

 space this indetermination can not be made 

 to disappear; the solution not becoming 

 determinate until the case of those bodies 

 in the plane is reached. 



Note on the Secidar Perturbations of the 

 Planets: Professor Asaph Hall, Pro- 

 fessor of Mathematics, U. S. Navy (re- 

 tired). 



It is known that the determination of the 

 secular perturbations of the principal 

 planets of our solar system depends on 

 the solution of an equation of the eighth 

 degree. The roots of this equation depend 

 on the masses of the planets; and if the 

 masses are changed the values of the roots 

 will change also. In this paper an example 

 is given of the changes in the roots, from 

 one set of masses to another, by means of 



the formulas computed by Stockwell. The 

 results indicate that the formulas of Stock- 

 well can be used with advantage, and that 

 the labor of solving the equation of the 

 eighth degree can be much diminished. 



The Bolyai Centenary: Professor G. B. 

 Halsted, Professor of Mathematics, Uni- 

 versity of Texas. 



On the fifteenth of December, 1902, is 

 the centenary of the discoverer of non- 

 Euclidean geometry, the Hungarian John 

 Bolyai, or, in Magyar, Bolyai Janos. This 

 extraordinarily important and suggestive 

 subject, non-Euclidean geometry, in its in- 

 ception, evolution, present state and near 

 future development, was treated in this 

 paper. 



The Approach of Comet b 1902 to the 

 Planet Mercury: Charles J. Ling, Man- 

 ual Training High School, Denver, Colo- 

 rado. 



The questions treated were: 

 The exact position of comet and planet 

 at time of nearest approach: to obtain ac- 

 curately distance between the bodies at 

 this time. 



The great velocity of comet near peri- 

 helion together with the position of orbit 

 takes the comet away from Mercury very 

 rapidly. The effect of Mercury at distance 

 of 2^ millions of miles very slight. 



Very questionable, if any, effect will be 

 produced by Mercury which will enable 

 astronomers to tell anything about mass of 

 Mercury. 



An Untried Method of Determining the 

 Constant of Refraction: George A. Hill, 

 U. S. Naval Observatory. 

 This paper called attention to a method 

 of deriving the constant of refraction from 

 transits of pairs of stars in the prime verti- 

 cal. Remarks were first made upon our 

 present knowledge of the constant as se- 

 cured from observations of stars at upper 



