108 RELATIONS BETWEEN SEVERAL [BOOK I. 



- - cotan * (N&quot; N} - \) cotan * (JT N) 



Hence, also, two other wholly analogous formulas are obtained by interchanging 

 the second place with the first or third. 



84. 



Since it is possible to determine the whole orbit by two radii vectores given 

 in magnitude and position together with one element of the orbit, the time also 

 in which the heavenly body moves from one radius vector to another, may be 

 determined, if we either neglect the mass of the body, or regard it as known : 

 we shall adhere to the former case, to which the latter is easily reduced. Hence, 

 inversely, it is apparent that two radii vectores given in magnitude and position, 

 together with the time in which the heavenly body describes the intermediate 

 space, determine the whole orbit. But this problem, to be considered among the 

 most important in the theory of the motions of the heavenly bodies, is not so 

 easily solved, since the expression of the time in terms of the elements is tran 

 scendental, and, moreover, very complicated. It is so much the more worthy of 

 being carefully investigated ; we hope, therefore, it will not be disagreeable to 

 the reader, that, besides the solution to be given hereafter, Avhich seems to leave 

 nothing further to be desired, we have thought proper to preserve also the one 

 of which we have made frequent use before the former suggested itself to me. 

 It is always profitable to approach the more difficult problems in several ways, 

 and not to despise the good although preferring the better. We begin with ex 

 plaining this older method. 



85. 



We will retain the symbols r, /, N, N , p, e, IT with the same meaning, with 

 which they have been taken above; we will denote the difference N --N by A, 

 and the time in which the heavenly body moves from the former place to the 



