160 
SECOND REPORT- 1832. 
is not assumed, (the methods being best adapted to the small 
planets,) and the tentative part of the operations differs from 
that commonly used in this respect, that two unknown quan¬ 
tities must be tried. In the Gottingen Transactions , tom. 2 , is 
a parabolic method by Gauss. In the Phil. Trans. 1814, 
Mr. Ivory gave a parabolic method, amounting to the same as 
Olbers’s. In the Berlin Ephemeris 1820, Olbers has given a 
method of correcting the approximate elements, and introducing 
the supposition of ellipticity; this, however, had been done by 
Laplace in the Mdcanique Celeste. In the Milan Ephemeris 
1817, and the Berlin Ephemeris 1824, Mosotti and Littrow 
have given methods. Pontecoulant has given a parabolic method 
in his Theorie Analytique. In the 5th book of the Mecanique 
Celeste , Laplace pointed out an alteration in his own method, 
and showed that the preliminary calculations (whose difficulty 
and inaccuracy had been considered the most formidable ob¬ 
jection,) might in fact be made very easy and accurate. La¬ 
grange left some remarks on the orbits of comets, which are 
published in the last edition of the Mecanique Analytique. 
In the Mem. Astr. Soc. vol. 4, Mr. Lubbock has shown that 
supposing the orbit to be parabolic, or supposing the major 
axis to be known, the equation may be reduced to a quadratic; 
and in a Supplement he has increased the accuracy of the me¬ 
thod, so as to make it applicable to observations at a greater 
interval. In this method, after an approximate determination 
of the orbit, on the supposition that it is parabolic, the major 
axis may be easily found, and may be applied to determine more 
exactly the orbit; in the present state of the science of comets, 
this is an important point. Finally, in the Berlin Ephemeris 
1883, Olbers has made some additions to his old method. 
All these methods (except Laplace’s,) require three complete 
observations, and can use no more ; and in every part of the 
calculations they require accurate numbers for those observa¬ 
tions, and calculation with 7-figure logarithms. Laplace’s can 
use any number of observations, and after the preliminary cal¬ 
culations requires no extreme accuracy in any part. The 
general methods (including Pontecoulant’s and Mr. Lubbock’s) 
fail when the apparent geocentric path passes nearly through 
the Sun’s place. 
The calculation of the true anomaly for a given time, by the 
common elliptic formulas, is troublesome and liable to error 
when the ellipse is very long. In the Monatliche Correspondent 
vol. 12, Bessel gave Tables for finding the true anomaly in a 
long ellipse ; Posselt, in vol. 5 of the Zeitschrift , has also given 
Tables. 
In the Mon. Corr. vol. 14, Gauss found apparently a short 
