g 



k 



Schubert's investigation of Kepler's Problem, 



# 



# 



r 



without recommencing the whole operation 



The method of La- 



place is, in fact, only an application of the analytical theory of 

 Functions, or'Taylor's famous Theorem, which is so useful in all 

 branches of mathematics. But, since the author of the Mecanique 

 Celeste has not gone through the whole calculus, and has, more- 



F 



over, treated the theory of Functions in a more general point of 



L 



view, than what this Problem requires ; it seems, that S. full and 



^ 



clear elucidation of this Problem will not be useless. Besides, 

 it will be found in the latter part of this paper, that I have 

 treated the subject in a manner, which is quite new. 



^ S. Let the semi-major axis of the elliptical orbit of a Planet 

 be equal to 1 ; 



the eccentricity e, the radi 



the 



the eccentrical, and the time 



hj, supposed to be counted 



from the Perihelium 



£ 



then the resolution of Kepler's Prob- 



lem will be contained in the three following equations 



L 







% 



e sm i 



III. % 



II. tana 



o 



V 



2 



tan^; 



1 



1 + e 



1 



e 



9 



e cos if 



by the help of which 



d 



b- 



be found from 



en 



o 



le 



giv 



/* 



When 



h 



other object 



calcul 



astronomical tables, the most natural and easy way is undoubt- 

 edly, first to find e from f^, by means of the equation I, for which 

 purpose there are several well known indirect methods ; and then, 

 io calculate directly v and % from e, by means of the equations II 



and 111 



But, in the calculus of the perturbation 



sary to develop u and 5; in series proceeding according to the 



powers of e and depending upon the angles ^, g^, Sfi, etc. 



these series is the real object of our problem 



Th 



gation of 



I shall begin with seeking v from 



by means of 



II, in order to know the form or nature of the function, to which 



the general theory of functions 



'; 



be applied 



in 



