2 THEORBITOFURANUS 



and mutual inclination are small, and has for that reason fallen, of late, into a 

 certain disrepute. The extended tables published by Le Vcrrier 1 have, however, 

 added so much to its facility for use that it is not wholly unworthy of attention. 



At the other extreme stands the purely mechanical method, in which special 

 values of the disturbing force are computed for many combinations of the mean 

 anomalies of the two planets, and the values of the coefficients in the general 

 expression for the force thence deduced. 



Between these two stands what I conceive we may designate as the Cauchy- 

 Hansen method, in which the development is made mechanically with respect to 

 the one planet, but the eccentric anomaly of the other is retained as an undeter- 

 mined quantity. The germ of this method is found in several papers, by Cauchy, 

 in the earlier volumes of the Comptes Rendus of the French Academy, which have 

 since been combined into a homogeneous memoir by Puiseux. 2 The object had in 

 view by these authors is only the computation of inequalities of long period. But 

 Hansen has taken up the essential principle of the method, first, in his prize memoir 

 on the perturbations of comets, crowned by the French Academy of Sciences, about 

 1848, and afterwards in his " Ameinandersetzung einer zweckmassigen Methode zur 

 Berechnung der Storungen der kleinen Planeten"* and applied it to the general 

 development of perturbations. 



Among the three methods of integration, the first in point of analytical elegance 

 and generality, but the last in order of convenience in use, is that of the variation 

 of elements, a method with which the name of La Grange is inseparably associated. 



In the second the direct integration of the differential equations which express 

 the perturbations of longitude, latitude, and radius vector is effected by special 

 devices. 



In the first of these methods the problem is presented in this form : The equations 

 of motion being completely integrated for the action of the principal forces only, 

 how must the arbitrary constants of integration vary in order that the same expres- 

 sions may represent the motion of the planet under the influence of the disturbing 

 forces'? In the second method, the same thing being, presupposed, the question is, 

 what expressions must be added to the integrals of undisturbed motion in order 

 that the sum may represent the integrals of the disturbed motion I 



The third is Hansen's method, in which the co-ordinates are partly expressed in 

 terms of a certain function of the time known as the disturbed mean anomaly, 

 determined by the condition that the true longitude in the disturbed orbit shall be 

 the same function of the disturbed time that the longitude in the elliptic orbit is 

 of the simple time. 



Although the last two methods have a great advantage over the first in the com- 

 putation of the periodic perturbations, I conceive the first to be best adapted to 

 the computation of the secular variations, and perhaps, of terms of very long period 

 in the mean longitude and the elements of the orbit. 



1 Annales de V Observatoire Imperial de Paris. Tome I. 



1 Annales de V Observatoire Imperial de Paris. Tome VII. 



Abhandlungen der Koniglich Sachsischen Oesellschaft der Wissenschaften. Band V. VI, VII. 



