determining the Orbits of Comets. 285 



V m XJt) + \di) * \~dt) ' 

 by which it assumes this form : 



in which likewise J, K, L are independent of r' and known ; 

 and combining this equation with the equation 



r' 3 = R' 2 - 2 p' R' cos (a' - e') + p' 2 sec 5 ' 2 , 



he finds p' and r' by trials. Their values substituted in the 



d «y' d if d z 

 expressions for — , --^-, -j-, give the elements, in as much 



as we know now the place in space by its three heliocentric 

 coordinates, as well as the projections of the linear velocity on 

 the three axes. For equal intervals of time the formula? are 

 rather more simple than for unequal ones, but the difference 

 in the convenience of the calculation is inconsiderable, and the 

 process in both cases the same. The method employs quan- 

 tities which are identical with, or analogous to, the numerator 

 and denominator of M, and is therefore not applicable to the 

 case of exception. The author calls, in this case, for other 

 observations. In order to judge of this method it is necessary 

 to investigate how many terms of the different series for x, 3/, z, 



dot} 

 x", y", #", as functions of #', -r- &c, have been taken into 



account. This may be most clearly understood by introducing 

 from the very beginning only so many terms as are unavoid- 

 ably required, but which at the same time are the only ones 

 that can be taken for obtaining the formulae of the author. 

 An examination thus conducted will prove that, supposing for 

 the terrestrial orbit 



dt 

 dY' 

 dt 



= -Y' 



= +X', 



which may be done, the formulae of the author will be com- 

 pletely obtained by taking the following terms, and no more. 



1 .. U X .I i (I J, 



„ d 1/ „ . dy' 



