LE VERRIER ON THE PERTURBATION'S OF PLANETS. 335 



the investigation which M. Binet presented in 1812 to the Aca- 

 demy of Sciences, and in which he states the error which had crept 

 into that part of the great inequaUty of Jupiter which depends 

 on the fifth order. But by the extent of that work, we may 

 easily judge that all hope of further advancing the approxima- 

 tions by this path must be lost. 



2. Interpolation seems then alone capable of furnishing the 

 coefficients corresponding to high multiples of the mean longi- 

 tudes. The calculations certainly are still very long ; but they 

 are not impracticable, like those which result from algebraic 

 developments. The disturbing function depends on the mean 

 longitudes of the disturbing planet and the disturbed planet ; 

 and these two longitudes, in the development of the function, 

 may be considered as independent variables. By attributing to 

 these variables particular values, we obtain numerical values of 

 the disturbing function ; a limited number of them serves for the 

 determination of a similar number of the coefficients of the de- 

 velopment effected according to the sines and cosines of the 

 multiples of the mean longitudes. 



We may employ, according to the well-known formulae, all 

 the numerical values of the function corresponding to mean lon- 

 gitudes equidistant from one another of an arc exactly dividing 

 the circumference. But this step is subject to an inconvenience 

 which cannot always be easily avoided. If, in fact, we know, 

 for a given number of values of the disturbing function, what is 

 the rank of the first term which is regarded as negligeable, fre- 

 quently nothing indicates whether this term is really small 

 enough to be neglected without altering the degree of accuracy 

 which it is important to obtain. And if we perceive, after ha- 

 ving effected the greater part of the calculations, that we should 

 have preserved only two terms more, we are obliged either im- 

 mediately to double the number of the numerical values em- 

 ployed, or to recommence the whole work, which will of ten pre- 

 sent less inconvenience. 



To avoid these difficulties, I propose the employment of a me- 

 thod of interpolation in which I satisfy the following condi- 

 tion : — 



Huvhiy already executed the calculations necessary for the de- 

 termination of vi of the coefficients, if v)e find that p others must 

 be preserved, thi<t may be done ivithont having executed more cal- 

 culations than if ire had had regard, from the beginning, to the 

 (n + p) coefficients. 



