CELESTIAL MECHANICS 389 



tion of constants. There was then for a time a pause in the develop- 

 ment of the theories of the major planets. 



The transition to the new century was, however, marked by the 

 beginning of a series of discoveries which are to-day still being 

 carried forward, and which in one direction have exerted an im- 

 portant influence in the development of celestial mechanics. I refer 

 to the discovery of the small planets. Laying aside the numerical 

 calculation of special perturbations, which has been developed to 

 a high degree of refinement, the perturbation problem will be con- 

 sidered here in the sense in which it- was brought over from the 

 preceding century. The interpolation formulae which the special 

 perturbations offer can permit only an extremely incomplete insight 

 into the nature of the motion. To be sure, the general perturbation 

 formulae in the form given by Laplace are also to be considered 

 merely as interpolation formulae, since they hold for only a limited 

 time, practically nothing being known in regard to the absolute con- 

 vergence of the series of secular terms. We shall first of all follow 

 the important investigations which have been made for the purpose 

 of representing the motion of a small planet by means of general 

 perturbation formulae in the sense spoken of. 



In addition to the Theoria Motus Corporum Coelestium, which, 

 for apparent reasons, does not here come under consideration, Gauss 

 busied himself with the theory of the minor planets, by making 

 extended investigations on the perturbations of Pallas. He did not 

 bring his work to a conclusion, and thus it has remained without 

 further significance. The prize problem given by the Paris Academy 

 in 1804, and repeated in following years, led in 1812 to the memoirs 

 of Burkhart and Binet which, however, inspired no further contri- 

 butions to the solution of the problem of the perturbations of the 

 small planets. Meanwhile an interesting comet was discovered in 

 1818 by Pons of Marseilles, whose orbit was computed by Encke, 

 and which was on that account called Encke's comet. The aphelion 

 of the comet lies within the orbit of Jupiter, the eccentricity is far 

 greater than that of any of the hitherto known planetary orbits, 

 and the inclination amounts to 12. If the formulae could be found 

 which represent the motion of this comet, the question in reference 

 to the small planets would also be solved. It was, however, not 

 merely from this point of view that Hansen set himself the problem 

 of obtaining such formulae. He doubted, in fact, the correctness of 

 the comet's acceleration found by Encke, and hoped by means of 

 general formulae to be able to settle that question. On the basis 

 of the differential equations given in the Fundamenta Theoria Or- 

 bitis quam perlustrat Luna, Hansen developed formulae for the per- 

 turbations of the logarithm of the radius vector, of the time, and of 

 the sine of the latitude. The essential difference from Laplace's 



