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L6 



Schuberfs investigation of ICephr^s Problem 



§ 17. By means of 



formulsE, it is easy to develop 



the series (A)^ or the true 



ly 



far as we pie 



tlie Proble 



b 



pletely resolved. But 



; and 

 whole 



operation is, strictly speaking, notliing but 



an indirect method of 



approximation ; and, as the discovery of the Binomial Theorem 

 would not be complete, if, we had not a general formula for each 

 Binomial Coefficient N,., independent of all others, so Kepler's 

 Problem cannot be said to be completely resolved, unless we have 



immediately, and independently 

 of the preceding or following terras, any term of v. Therefore, 

 knowing that w is a series composed of terniis of this form Ae" 

 sin itifif the direct solution o 



a general formula, which gives 



f our 



Problem is reduced to the in- 



vestigation of a general formula, by which the coefficient or 

 multiplicator A may be found immediately, and without having 

 recourse to the above series, ?, sin i «, x*, whatever may be the 



numbers a and 



other words, to 



A by 



a 



ex- 



function of a and m only. Such a general and independent 

 pression of A is particularly useful, when the object is, to verify 

 an isolated term which, for some reason, appears to be doubtful ; 

 a case by no means rare, as every one knows, who is acquainted 

 with these calculations. Moreover, it will appear, that the com- 

 putation by this method is easier than by the common one, and it is 

 not so liable to mistakes in the calculation : it shews also the sen- 

 eral law, by which all the terms of v are formed, whereas this 

 law is quite lost in the common method. 



§ 18. The Problem, which we have now to resol 



hat 



may be the numbers a and m. in the 



Ac" sin mu* to find 



general expression of A. But there is a very essential obsei 



on to be made, viz, that the numbers a and m are not wholly 



