determining the Orbits of Comets* 1 9 



where 



m = -r|£ , log m s 0*9862673, 



If we make use of the table given at the end, u must be so 

 determined that if 



then 



(r + f"j* 



where log 2 K = 8-5366114. 



In making these trials hitherto, a value of u was assumed 

 and from it r, r", k were computed. The substitution of the 

 values thus found in Lambert's formula showed whether the 

 assumed value of u was the true one. If not, it was varied 

 until the formula was complied with. But this way of pro- 

 ceeding has this disadvantage, that in general ti 9 according to 

 its geometrical signification, does not admit of an approximate 

 valuation, and one might probably assume at first a value 

 very far from the truth. The process proposed by Olbers 

 in the preceding paper appears to be much preferable. 

 From practical, and, as far as here applicable, from theo- 

 retical reasons, it will be perceived that r + r" will be sel- 

 dom or never Z. 1, and likewise seldom 7 3. The value 

 r + r' f = 2 will, at least in most cases, not be far from the 

 truth. If, therefore, by means of the table, from r -f r" = 2, 

 the corresponding k is determined, and from this value of k 

 the u, and thence r and /*", the comparison of the new value of 

 r + r" with the one assumed for this sum, will prove how far 

 the assumption was correct. The new value may now be again 

 applied, and we may thus continue until a perfect agreement 

 is obtained. 



The formulae being so simple, it is interesting more closely 

 to investigate the steps of this process. The factor \l may be 

 considered as a constant quantity, it being at any rate, in all 

 cases which can occur, little different from unity. Making, 

 for brevity, r -f r" = s, the differential equations of the for- 

 mulae, as they are successively applied, will give the following 

 relations : 



dk= -i# — 



2 s 



du = — d k 

 u 



D2 



