4; 



Schubert's investigation of Kepler's Problem, 



£ 2 



€ + S A sin . + -\^ sin § e + + - \' sin i e ; 



2 - t 



the letter i 



denoting every integer affirmative number. 



§ 4. Since tlie eciuation (A.) gives v in e and sin le, our prob- 

 lem is now reduced to this, to develop e as well as sin it in series 



proceeding according to the powers of e, which must he perform- 

 ed by the help of the equation I. If, therefore, as is the case in 

 our problem, ^ is given, there will be found in every planetary or- 

 bit, according to the greater or less degree of its eccentricity, dif- 



same given ang 



le 



i" 



fereut values of e, corresponding to the 



Whence it follows, that /n is to be regarded as a constant or given 



quantity, and e as a function of 



If we then put e 



X 



} 



i 



ij, y will be a function of x, and the equation I will have the 



following form 



iuvesti 



y 



y 



by means of 



are to 



ate y, or sin iy, or any other function of y. Moreover sin y 

 being also a function of y, the equation 1 agrees with the more gen- 



L 



eral one. Ml o 



yxx.<^{y) 



by which ^f (y) is to be devel 



oped 



proceeding according to the powers of a? ; f (y) 



« 



and I (y) being any two functions of ^, instead of which we shall 

 write, for the sake of brevity, <p and 4/. 



§ 5. However difficult the general solution of the equation (1) 

 may be, on account of the nature of the function ©, there will al- 



particular cases, wher 



for instance, supposi 



difficulty 



3 



X 



find immediately y 



a 



consequently 4^ (y) 



9 



I 



(a). But as soon as it is kn 



I 



particular case in which 



shall 



h, 4/ [y) becomes equal to 



We 



every other case, in which 



5 + A, accordin 



Taylor's Theorem 



# 



^(y) 





dx 



1.2'da;3 



« « • • 



+ 



/t" 



4- 



1.3,..?j * d,T«' 



■'■- 



