160 Afttronomical and Naiitiral Collections. 



grange has attempted to show, of a higher order, unless we 

 admit a supposition, which, whether justly or not, he thinks 

 objectionable. Boscovich, proceeding upon the same supposi- 

 tions which he employs in his construction, has obtained an 

 equation of the sixth degree, which affords a very correct ap- 

 proximation, if the observations are so accurate that they may 

 be employed at short intervals of time. Lambert's second me- 

 thod is founded on some acute considerations on the apparent 

 path of the comet ; but it is wholly useless ; at least neither Pin- 

 gre nor myself could succeed in employing it with advantage, 

 partly because great accuracy is required in the observations, 

 and partly, because the suppositions concerned are not quite 

 correct ; at the same time it may sometimes be of use to apply 

 the interesting theorem respecting the deviation of the apparent 

 path of the comet from a great circle, to the purpose of decid- 

 ing whether the comet is nearer to the sun than the earth, or 

 not. The prize proposed by the academy of Berlin for the so- 

 lution of the problem was adjudged to Mr. Von Tempelhof, and 

 to M. De Condorcet ; and the accessif, or second premium, to 

 Mr. Hennert. I confess that I am not sufficiently acquainted with 

 all these solutions, but I do not find that they have been much 

 employed by practical astronomers. But the occasion seems to 

 have excited the very valuable rival researches of Lagrange, 

 Dusejour, and Laplace. Lagrange has given us three solutions 

 of the problem, all depending on equations of the sixth, or of a 

 still higher degree. The first of these he seems afterwards to 

 have thought imperfect ; indeed, Laplace has discovered a 

 slight error in the calculation, and Pingre was unsuccessful in 

 an attempt to apply it to practice. The second requires six 

 observations, which must be near each other in pairs, and it leads 

 by means of intricate calculations, to an equation of the sixth 

 degree ; it may, however, be sometimes of use, and Schulze 

 has determined the orbit of the comet ^of 1774 by it with tole- 

 rable accuracy. The third, which is calculated to excite the 

 highest admiration for the refinement of the analytical powers 

 which it exhibits, requires a very laborious preliminary calcula- 

 tion, and then the solution of an equation of the seventh or 



