526 Royal Astronomical Society. 



gent, whereby a smaller number of terms is required to be computed, 

 but the coefficients of the individual terms are obtained with a smaller 

 amount of labour than was necessary in the methods hitherto em- 

 ployed. 



It will be readily seen from whathas now been said, that the general 

 aim of M. Hansen's researches is the improvement of the methods 

 of expressing the lunar and planetary perturbations, so as to render 

 the numerical calculations more easy and more certain. There is, 

 however, one advantage which M. Hansen states to belong to his 

 method, of by far too important a kind to be passed over without 

 particular notice. It is this : — In the series which express the values 

 of the disturbed coordinates, every term exceeding a certain nu- 

 merical value, assumed at pleasure, can be immediately recognised, 

 so that all those terms which fall below the assumed value may be 

 rejected from the first, with the certainty that their sum falls within 

 a given limit. The certainty thus acquired that every term having 

 a sensible value is retained in the calculation, is an improvement 

 in the theory on which it would be difficult to set too high a value ; 

 and in fact it removes the principal defect which has hitherto at- 

 tended all the methods of approximation which have been proposed. 

 Nor is this advantage obtained by any sacrifice of generality ; for 

 neither with respect to the eccentricity and inclination, nor powers 

 of the mass, is any other restriction introduced than is inseparable 

 from the nature of the problem. 



Besides these principal advantages of more rapidly converging 

 series, and certainly with respect to the value of the omitted terms, 

 there are some minor advantages incidental to the new method, 

 which, however, are still of great importance. Among these may 

 be mentioned certain relations subsisting among the analytical ex- 

 pressions of the coordinates, pointed out by M. Hansen, from which 

 equations of condition are deduced which not only facilitate the cal- 

 culations but afford a ready means of verification. 



The applications which M. Hansen has as yet made of his me- 

 thod are to the inequalities of Jupiter and Saturn*, in a memoir 

 which obtained the prize of the Royal Academy of Sciences of Berlin ; 

 and, to the lunar theory, in a work recently publishedf. In the 

 former memoir the theory is worked out to a numerical result. The 

 expressions for the differential values of the longitude, latitude, and 

 radius vector, are integrated by the method of quadratures, and re- 

 sults obtained which agree with those derived from the ordinary 

 methods of approximation founded on the smallness of the eccen- 

 tricities and inclinations. The approximations are, indeed, only 

 carried to terms of the second order inclusive, with respect to the 

 masses ; but in the case of Saturn, all the terms of this order ex- 

 ceeding a certain numerical value are calculated. His theory of the 



* Untersuchung ubcr die gegenseitigcn Storungen des Jupitcrs und Saturns. 

 Berlin, 1831. 



t Fundamenta nova Investigationis Orbitcc vera quam Luna perlustrat. 

 Gothse, 1838. 



