CELESTIAL MECHANICS 391 



of the small planets. Meanwhile all the calculations made in this 

 department of celestial mechanics soon showed that the path laid 

 out by Hansen does not lead to the object desired. Above all, with- 

 out an immense expenditure of time and labor no trustworthy re- 

 sults can be obtained for planets that occur in the neighborhood 

 of the so-called gaps, for which the terms of long period cannot be 

 accurately determined; and besides, in this case, the convergences 

 of the secular terms are much slower. All attempts in this direction 

 lead only to the result that at best we may obtain in this way 

 approximate perturbation formulae which for a considerable time will 

 guarantee the rediscovery of the planet, without claiming to represent 

 the observations. 



The circumstance that a large portion of the small planets occur 

 in the neighborhood of the so-called gaps, thus causing such an 

 increase in the perturbations that after a relatively short time these 

 can no longer be considered as small quantities, led Gylde"n to state 

 the question in the following manner: "Will it be possible to deter- 

 mine the elements as absolute constants, and to so determine the 

 terms of long periods (thus avoiding completely the introduction 

 of the time explicitly) that the intermediate orbit thus obtained 

 shall remain included within definite limits, and only differ from the 

 real orbit by quantities of the order of the masses of the planets?" 

 This question includes the question of stability, and the principal 

 problem thus consists in proving the convergence of the long-period 

 series. Gykten believed that he could establish the convergence by 

 means of what he called the horistic method. Poincare", however, 

 disputes the correctness of this method. On this assumption Gyl- 

 deVs theory would be merely an hypothesis. Even if the method is 

 correct, it is applicable only with reference to a limited number of 

 small planets, as it is based upon the development in powers of the 

 eccentricities and inclinations of the disturbing and disturbed planets. 

 Here we stand, so far as this question is concerned, at the end of 

 the nineteenth century. Upon the problem presented at the begin- 

 ning of the century much skill and labor has been spent; a satis- 

 factory solution has not, however, been reached. 



If now we turn to the larger planets, a more gratifying picture 

 presents itself. Here we find at the end of a century a work well 

 completed; taking it all in all, theory has in this case mastered 

 quite satisfactorily the century's immensely rich and abundant ob- 

 servations. 



Not only the number of observations made during the first half 

 of the century, but still more their epoch-making precision, which 

 is linked with the names of Bessel and Struve, soon showed that the 

 numerical formulas of Laplace were not sufficient to satisfy the 

 increasingly accurate observations. In individual cases, it is true, 



