12 RELATIONS PERTAINING SIMPLY [BOOK I. 



11. 



The inverse problem, celebrated under the title of Kepler s problem, that of 

 finding the true anomaly and the radius vector from the mean anomaly, is much 

 more frequently used. Astronomers are in the habit of putting the equation of 

 the centre in the form of an infinite series proceeding according to the sines of the 

 angles M, 2M, BM, etc., each one of the coefficients of these sines being a series 

 extending to infinity according to the powers of the eccentricity. We have con 

 sidered it the less necessary to dwell upon this formula for the equation of the 

 centre, which several authors have developed, because, in our opinion, it is by 

 no means so well suited to practical use, especially should the eccentricity not be 

 very small, as the indirect method, which, therefore, we will explain somewhat 

 more at length in that form which appears to us most convenient. 



Equation XII, E = M-\- esmfi, which is to be referred to the class of tran 

 scendental equations, and admits of no solution by means of direct and complete 

 methods, must be solved by trial, beginning with any approximate value ofJE, which 

 is corrected by suitable methods repeated often enough to satisfy the preceding 

 equation, that is, either with all the accuracy the tables of sines admit, or at least 

 with sufficient accuracy for the end in view. If now, these corrections are intro 

 duced, not at random, but according to a safe and established rule, there is scarcely 

 any essential distinction between such an indirect method and the solution by 

 series, except that in the former the first value of the unknown quantity is in a 

 measure arbitrary, which is rather to be considered an advantage since a value 

 suitably chosen allows the corrections to be made with remarkable rapidity. Let 

 us suppose t to be an approximate value of E, and x expressed in seconds the cor 

 rection to be added to it, of such a value as will satisfy our equation .&quot;= t -j- x. 

 Let e sin e, in seconds, be computed by logarithms, and when this is done, let the 

 change of the log sin e for the change of 1&quot; in e itself be taken from the tables ; 

 and also the variation of log e sin e for the change of a unit in the number e sin e ; 

 let these changes, without regard to signs, be respectively A., p, in which it is 

 hardly necessary to remark that both logarithms are presumed to contain an 

 equal number of decimals. Now, if e approaches so near the correct value of E 



