Olbers's Method of determining the Orbits of Comets. 281 



determination of the orbits of comets. If, therefore, in the new 

 representation of it, the steps and the precepts of Laplace had 

 been exactly retained, a further explanation would be super- 

 fluous. But on the very points in which M. de Pontecoulant 

 felt himself obliged to deviate from his illustrious model, the 

 above-mentioned opinion is founded, and these points likewise 

 supply us with the reasons for which we must withhold our 

 assent. It is well known that Laplace sets out from the gene- 

 ral equations of motion, and introduces, in place of the second 

 differentials of the coordinates, the first and second differen- 

 tials with respect to time of the observed longitude and lati- 



'j ,. . . da' d 2 u' d# tPt''. 



tude, or, according to our notation, ~r- 9 -=—£, -r— , — -. 



CL t (L l CI I OL t" 



The method would be perfectly strict if these latter quantities 

 could be accurately determined. But as there are no other 

 means of finding them than the first and second differences of 

 the observed quantities, Laplace expressly remarks, (Mec. Cel., 

 torn. i. p. 203,) that the observations must be selected and mul- 

 tiplied in order to obtain the data as accurately as possible; 

 and although he gives himself a method of approximating the 

 truth from three observations only as nearly as possible, but 

 certainly in a circuitous manner, still he says, at the conclusion 

 of it (Mec. CeL, torn. i. p. 211), that it would be more simple 

 and more accurate to make use of more than three observa- 

 tions. 



In contradiction to this M. de Pontecoulant observes that 

 it has been found by experience that the accuracy is not greater 

 if more than three observations are used, because in this case 

 the errors of observation have greater influence in propor- 

 tion to the number employed. Practically considered this 

 certainly cannot be admitted. If it is at all possible to de- 

 rive a quantity from observations, it must be the more accu- 

 rately determined the more data are employed ; and should 

 this not be the case, there is at least sufficient reason for be- 

 lieving that from the less number it can only be very inaccu- 

 rately determined. Moreover, this remark as well as the 

 whole discourse seem to involve the belief that errors of obser- 

 vation only have an influence on the more or less accurate de- 

 termination of the differential quotients. But in accordance 

 with Laplace, (Mec. CeL, tom.i. p. 257,) one may easily convince 

 one's self that even in the case of absolutely accurate observa- 

 tions the determination of those is always approximate, and 

 that consequently the errors of observation contribute indeed 

 toward the uncertainty, but are by no means the sole nor even 

 the principal cause of it. 



Third Series. Vol. 7. No. 40. Oct. 1835. 2 O 



