390 ASTROMETRY 



theory consists in the fact that Hansen employs arguments contain- 

 ing multiples of the eccentric anomaly of the comet, of the mean 

 anomaly of the disturbing body. To a given finite power of the ratio 

 of the two semi-major axes there thus belongs a double series, which, 

 with reference to the disturbing body, is an infinite series of powers 

 of the eccentricity, but with reference to the disturbed body is a 

 finite series. This theory Hansen published shortly after 1830 under 

 the title of Storungen in Ellipsen von grosser Excentricitat, and at 

 the same time made an attempt to obtain the general perturba- 

 tions of Encke's comet produced by Saturn. Although the com-' 

 putations were not brought to a definite close, still it cannot be 

 doubted that his method is useful for this case. As the perturbations 

 by Jupiter are far more important, both with reference to Encke's 

 comet and to the small planets, and cannot be obtained by this 

 method, Hansen's work cannot be considered as satisfactory, but 

 rather as a failure, at least with reference to Encke's comet. He, 

 therefore, attacked the problem from an entirely different stand- 

 point, and devised the so-called partition method, which he pub- 

 lished in his Paris prize memoir, together with an application to 

 the perturbations produced on Encke's comet by the planet Jupi- 

 ter. This example was also not carried to an end, evidently for the 

 simple reason that this was practically impossible, and thus we see 

 that this method also was unable to solve the problem. 



After his unsuccessful effort to obtain general perturbations for 

 such eccentric orbits as that of Encke's comet, Hansen turned his at- 

 tention especially to the small planets, and by a further development 

 of the method given in his first memoir succeeded in giving formula 

 by means of which he was enabled to represent the motion of the 

 planet Egeria, at least for the time embraced by the observations 

 at his disposal. Unfortunately in this, as in the case of so many 

 other small planets, theory and observation deviated more and 

 more the more distant the latter lay from the epoch of the former, 

 so that after about fifty years, his tables no longer satisfactorily 

 represent the observations. Later several of Hansen's pupils, Les- 

 ser, Blecker, and others, computed the general perturbations of 

 some of the small planets. The most prominent of Hansen's pupils, 

 Gylde"n, again took up Hansen's partition method and substituted the 

 mean anomaly of the planet and the partial anomaly of the comet 

 by means of elliptic integrals, and thus obtained a much greater 

 convergence in the development of the perturbative function. In 

 the determination of the constants of integration, the old difficulties 

 reappeared, so that taken as a whole no success appears to have been 

 gained. Gylde"n sought further, by means of a skillful combination 

 of Hansen's partition method with a special development of series, to 

 obtain a simpler method for the computation of the perturbations. 



