334 



Article VIT. 



0)1 Interpolation applied to the Calculation of the Coefficients of 

 the Development of the disturbing Function. By U.-J. Le 



Verrier. 



[From a separate Treatise. Paris, 1841.] 



1 . JL HE determination of the periodical and secular inequali- 

 ties of the planets is carried back, by the theory of the variations 

 of the arbitrary constants, to the investigation of the develop- 

 ment of certain expressions which are functions of the time and 

 elements of the orbits. These functions are reduced into a series 

 proceeding according to the sines and the cosines of the different 

 multiples of the mean longitudes. And when the numerical va- 

 lues of the coefficients of the principal terms of these series have 

 been calculated, we easily arrive at the knowledge of the pertur- 

 bations themselves of the planets. 



To obtain one of the coefficients in particular, the Mecanique 

 Celeste supposes that we commence by forming its analytical ex- 

 pression in function of the disturbing mass, of the semi-major 

 axes, of the excentricities and the inclinations of the orbits of the 

 two planets under consideration, in function of the longitudes of 

 their perihelia and their nodes. This algebraical development, 

 which rests wholly on the employment of Taylor's theorem, offers 

 no other difficulty than the length of the literal calculations. 

 But this difficulty is immense. Thus, notwithstanding all the 

 care of Burckhardt, the analytical expression which he deter- 

 mined for that part of the great inequality of Jupiter and Saturn, 

 which depends on the fifth powers of the excentricities and in- 

 clinations, was found to contain some inaccuracies. Thus Mr. 

 Airy, to obtain the expression of the inequality of a long period 

 which Venus introduces into the mean movement of the Earth, 

 had to go through a very long process ; and other geometers, 

 starting from the same data, have been unable to find again ex- 

 actly the same results. This inequality is however only of the 

 fifth order. To what difficult labour should we then be led by 

 the method of the algebraic developments, if we recognized the 

 necessity of having regard, in some theories, to inequalities of a 

 higher order? 



We might, it is true, attain to the seventh order, by means of 



