determining the Orbits of Comets, 283 



cally considered, setting aside all errors of observation, Olbers's 

 method is more accurate by a whole order, as it neglects quan- 

 tities of the second order only when the intervals of time are 

 unequal ; but when these are equal, it takes even those quan- 

 tities into account. Here we have most likely the reason of 

 the remark so frequently made in former times, that Laplace's 

 method leads much more slowly, than the calculators wished, 

 to a sufficiently accurate approximation: for the somewhat 

 tedious preparatory calculation for a great number of obser- 

 vations and the wish to obtain as soon as possible an approxi- 

 mate knowledge of the orbit, generally induced calculators to 

 use three observations only. Although the errors of obser- 

 vation must have a greater influence on Laplace's method, 

 because the second differences of the observations are imme- 

 diately employed, yet this greater influence alone would, in 

 modern times, not be a motive for abandoning it, as the possi- 

 ble errors are now so much diminished. 



M. de Pontecoulant observes, that there are cases where, 

 to the exclusion of every other, Laplace's method must be 

 applied *. Such cases are not known to me. Should he re- 

 fer to the above-mentioned case of exception, it is clear that 

 Olbers's method likewise answers the purpose. The course 

 of proceeding is then analogous in both methods. By making 

 use of a coefficient very similar to the M above used, Laplace 

 reduces the problem to the ascertaining of two quantities, viz. 

 r' and p' 9 from two equations, one of the second, the other of 

 the third degree. When this coefficient is too indeterminate, 



dp' 

 another unknown quantity,-^-, is introduced, and the problem 



is reduced to the determination of three unknown quantities 

 from three equations of the second, third, and fourth degree. 



The above-mentioned coefficient gives -~ as a pure func- 



tion of p' 9 just as M serves for determining p" from p 9 and by 



it the quantity f — ,, — ^mr ) ls eliminated, which, consequently, 



when the coefficient cannot be applied, reenters into the equa- 

 tion. The difference of both methods consists, therefore, in 

 the cases of exception in Laplace's introducing the quantity 



\~/3~"p73") mto tne fi fia l equations, as a new determining 



* " II y a meme des cas ou il est indispensable de V employer" — torn, ii. 

 p. 486. 



2 02 



