Januabt 15, 1909] 



SCIENCE 



85 



force. Hill proposed to take as this first 

 approximation an orbit which would in- 

 clude all the inequalities depending upon 

 the mean motions of the sun and moon. 

 The old theories consisted essentially in 

 suitably varying a solution of the problem 

 of two bodies, while Hill's theory seeks 

 the true orbit by attempting to vary ap- 

 propriately the restricted problem of three 

 bodies. During the last fifteen years, 

 Brown has published a series of papers, 

 concluding with the 1907 Adams Prize 

 Essay of the University of Cambridge, 

 which extend Hill's work to the construc- 

 tion of the most perfect of all the ten or 

 eleven theories of the moon which have 

 appeared since Newton's "Principia." 

 Hill found periodic solutions of the mo- 

 tion of a particle in a plane under the in- 

 fluence of two bodies which revolve round 

 each other in circular orbits and whose 

 distance apart is infinite. In its initial 

 stages Brown 's theory modified Hill 's solu- 

 tion in two particulars, first by reducing 

 the distance of the two bodies to finite 

 dimensions, and thus introdiicing the in- 

 equalities which involve the solar parallax, 

 and second by including those inequalities 

 which are due to the moon's eccentricity. 

 Adequate accounts of these theories are 

 given in the presidential addresses de- 

 livered on the occasions of the award of 

 the gold medal of the Royal Astronomical 

 Society to Hill in 1887, and to Brown in 

 1907, while the relations of Brown's per- 

 fected work to the highly original pioneer 

 work of Hill are exhibited in the introduc- 

 tion which Poincare has written to Hill's 

 "Collected Works." Brown has recently 

 finished his complete numerical theory, and 

 lunar tables based upon it are to be pub- 

 lished by Yale University. His numerical 

 results furnish an interesting confirmation 

 of the validity of Newton 's law. Newcomb 

 proposed an explanation of the motion of 

 Mercury's perihelion by changing the ex- 



ponent 2 in the Newtonian law to 

 2 + 0.00000016. Brown finds in his theory 

 of the moon's motion that the exponent can 

 differ from 2 only by ± 0.00000004. 



The work reviewed up to this point in 

 our discussion has found its sources in 

 Hill's periodic solution, the memoir of 

 Lagrange, the non-existence theorems of 

 Bruns and Poincare, and Lie's theory of 

 contact transformations ; that which follows 

 may trace its origins to Poincare 's theo- 

 retical and Darwin's numerical investiga- 

 tions on periodic solutions, Newcomb 's and 

 Lindstedt's solutions in trigonometric 

 series, Gylden's theory of absolute orbits, 

 and Painleve's theorems on the singulari- 

 ties of the problem. 



Although periodic and asymptotic solu- 

 tions do not exist in nature their services 

 to astronomy have been two-fold: to the 

 practical astronomer in supplying first ap- 

 proximations to orbits under investigation, 

 and to the mathematical astronomer in 

 opening the way to further theoretical re- 

 searches through what Poincare has char- 

 acterized as "la seule breche par" on nous 

 puissons essayer de penetrer dans une place 

 jusqu'ici reputee inabordable. " Darwin 

 has constructed a splendid collection of 

 examples of these orbits, planetary and 

 lunar; among his most curious satellite 

 orbits are perhaps those which present 

 three new moons in a month, and another 

 which has five full moons in one period. 

 Darwin 's orbits were subjected to a search- 

 ing analytical examination by Poincare 

 who showed that two sets of curves which 

 Darwin treated as continuous can not be 

 considered as such ; the true sequence of the 

 orbits in question has been exhibited by 

 Hough. Certain of Darwin's results have 

 been derived analytically by Charlier, and 

 specially with reference to the families of 

 oscillating satellites in the vicinity of the 

 five centers of libration corresponding to 

 the exact Lagrangian solutions. In Char- 



