iv INTRODUCTION. 



4. We see from this, that all values of ¥f and its derivatives, any one of which 

 may be denoted by /"(a), will depend upon a series of the form 



(2) f(*) = A t a t + A l a i +* + A t a i + l + A s a t + ' + A i a i + ' + '---, 



hi which A , A 1} A 2 , &c. are known functions of s and i; and to obtain the 

 values of b li J and its derivatives hi the mutual action of any two planets, it is only 

 necessary to substitute the proper value of a, and sum the series to as many terms 

 as will give them with the requisite degree of approximation. 



5. It is obvious, that not only must these series converge, which they all do, 

 though in very different degrees, but the facility with which they can be used must 

 depend very much upon the degree of convergency. Now it is found, that, although 

 all the values of a fall between and 0.75, these series, with but few exceptions, 

 converge so slowly that they are nearly useless in their present form ; and the great 

 problem has been to transform them into others converging more rapidly. 



This problem Leverrier has treated with ability and success, in the paper to which 

 reference has been made. The coefficients in his transformed series have received 

 the name of the Leverrier Coefficients. 



6. At the request of the Superintendent of the American Ephemeris and Nauti- 

 cal Almanac, the values of the Leverrier Coefficients were computed by the late 

 Sears C. Walker, assisted by Mr. Pourtales, and are published in an Appendix to 

 that work 1 for 1857. This carefully prepared paper has been of great aid to me, 

 especially the manuscript sheets containing the numerical values of the coefficients 

 carried to a high degree of accuracy, which were better adapted to the changes 

 demanded by the form of the following Tables. 



7. Upon undertaking this work, the first question which presented itself was, 

 How do the series giving the values of ¥% a D x b®, a? D\ ¥% &c, vary with a "? 

 Now, for special values of s and i, these coefficients are simply functions of a; and, 

 if their variation with reference to a is slow, terminating in low orders of differ- 

 ences, we may not only make this circumstance a check upon the accuracy of the 

 work, if we compute them for equidistant values of a, but we may tabulate them 

 with reference to a as an argument, and afterwards enter these tables with the 

 special values of a for the system, and take out their corresponding values. If 

 these Tables were extended from a = to a = 0.75, we should include all the 

 three hundred and sixty-four values of a; and, if the argument intervals were 

 sufficiently small, all the coefficients could be taken from them with trifling labor. 

 Besides, they would undoubtedly include all the planets hereafter to be discovered. 



It was soon found, however, that these variations, instead of being slow, were hi 

 most cases so rapid as to make them entirely useless, with any reasonable amount 

 of labor, for the purposes hidicated. 



8. But, resuming the equation 



f(a) = A a 1 + A, a i + 2 + A % a i + i + A 3 a ; + 6 + A, a j + 8 + ■ • • • , 



may we not find 



f{a) = f' (a) (a transformed series), 



in which /' (a) is an exact function of a, involving nearly the whole variation of 



