Januabt 15, 1909] 



SCIENGM 



89 



and coordinates ; these series, botli in their 

 terms and in their coefficients, are built 

 up from certain developments which 

 Bohlin has derived for roots of the funda- 

 mental quintic met with in Lagrange's 

 problem of the mutual distances. 



The question of the validity of certain 

 methods of Gylden has been the source of 

 considerable discussion among mathemat- 

 ical astronomers during the period under 

 review. The appearance of a long memoir 

 by Buchholz on Gylden 's horistie method, 

 and its convergence, brought forth from 

 Backlund a protest against the manner in 

 which the material of the memoir had been 

 accumulated and presented. To this pro- 

 test Buchholz replied with a defense of 

 the course he had pursued in preparing the 

 work; and a little later he published 

 another note objecting to a statement by 

 Schwarzschild that Poincare, in his prize 

 memoir, had proved the divergency of the 

 series employed by astronomers. About 

 this time Poincare examined in detail the 

 second of Gylden 's two horistie methods, 

 the first being open to grave objections as 

 had been shown by himself and Backlund. 

 As a result of his investigation Poincare 

 found that the second method, conveniently 

 modified, is a legitimate one, not for the 

 search of the general solution, but for the 

 determination of one of those particular 

 solutions which he himself had termed 

 periodic. He pronounced futile the effort 

 to derive from the horistie method develop- 

 ments uniformly convergent in the geo- 

 metric sense of the word, and declared 

 false Gylden 's conclusion that the terms 

 of high order in the perturbative function 

 can never produce libration. Poincare 's 

 results were questioned by Backlund and 

 an interesting controversy ensued, some 

 points of which were elaborated upon in a 

 later extensive memoir which Poincare de- 

 voted to Gylden 's theory, where he pointed 

 out gylden 's great service to science in 



creating a number of new methods which 

 have been applied with success, to certain 

 problems of mathematical astronomy, as for 

 instance, in the theory of the small planets 

 developed by Harzer and Brendel. He 

 found the methods proposed in Gylden 'a 

 earlier memoirs to be correct in the main, 

 but possessed of little more than historic 

 interest, having been superseded by less in- 

 convenient methods such as those of HiU 

 and Brown. Gylden 's later theories Poin- 

 care subjected page by page to a searching 

 critical examination which resulted in a 

 declaration that they are invalidated 

 throughout by errors arising in the initial 

 stages of Gylden 's analysis. 



Thanks to the recent researches of Levi- 

 Civita, Bisconcini, Sundman and Block, in-i 

 spired as they were by an earlier theorem 

 of Painleve the qualitative solution has 

 been attained in the field of the formal 

 resolution of the mathematical problem of 

 three bodies, and some progress has been 

 made towards the same end in the astron- 

 omical problem. Painleve demonstrated 

 that starting from given initial conditions 

 singularities occur only if one at least of 

 the mutual distances tends towards zero, 

 when t converges to a finite value *i. 

 When these singularities have been located, 

 the recent theorems of Mittag-LefSer, on 

 the representation of monogenic branches 

 of analytical functions, warrant the as- 

 sertion that the coordinates are expressible 

 in every case, and throughout the duration 

 of the motion, in series possessing the 

 fundamental properties of Taylor's series. 

 It may be remarked in passing that Vol- 

 terra has given examples of the applica- 

 bility of Mittag-Leffler's developments to 

 certain cases of the general %-body prob- 

 lem. Prom the standpoint of the qualita- 

 tive resolution of the problem, it becomes of 

 paramount importance, then, to define with 

 precision the initial conditions which lead 

 to a collision. Painleve in his Stockholm 



