46] ROYAL ASTRONOMICAL SOCIETY TO M. LE VERRIER. 365 



Fortunately, however, these are not needed for the actual requirements 

 of Astronomy, and for any definite period the simultaneous integrals may be 

 determined with any degree of accuracy that may be desired by the method 

 of quadratures. 



In this way M. Le Verrier has determined the values of the elements 

 for a period of 2000 years, starting from 1850, at successive intervals of 500 

 years. The first steps in this integration were attended with some difficulties, 

 because the determination of the numerical values of the rates of change of 

 the several elements at the various epochs depends on the elements them- 

 selves which are to be determined. Hence several approximations were 

 necessary in order to obtain the requisite precision. 



After this work of M. Le Verrier, however, the extension of the investi- 

 gation to other epochs, past or future, is no longer attended with the same 

 difficulties. In fact, from his results we may at once find, by the method 

 of differences, very approximate values of the elements at an epoch 500 

 years earlier or later than those which he has considered. His general 

 formulae will then give the rates of change of the several elements at the 

 epoch in question, and having these we can determine by a direct calculation 

 the small corrections which should be applied to the approximate values of 

 the elements first found. 



This process may evidently be repeated as often as we choose. 



It is important to remark that in the formula which give the rates of 

 change of each of the elements at the five principal epochs considered, as 

 well as in those which give the total variations of the elements at the same 

 epochs, the masses of the several planets appear in an indeterminate form, 

 so that it may be at once seen what part of the variation of any element 

 is due to the action of each of the planets, and what changes would be 

 produced in the value of any element at any epoch by any changes in the 

 assumed values of the masses. 



Consequently, when the astronomer of the future, say of 2000 years 

 hence, has determined the values of the elements of the planetary orbits 

 corresponding to that epoch, it will be easy for him, by comparing those 

 values with the general expressions given by M. Le Verrier, to determine 

 with the greatest precision the actual values of the masses, provided that 

 all the disturbing bodies are known ; and should there be any unknown 

 disturbing causes, their existence would be indicated by the inconsistency of 



