;1S() Aatronomicul mul l^aulical Collections. 



§. 76. 

 Besides these two methods of correction, I shall explain ano- 

 ther which really seems to be the most commodious, where the 

 elements of a parabolic orbit only are in question : and even if 

 it were less advantageous, it would be of use to be in possession 

 of a variety ; especially as the two former methods sometimes 

 fail. Laplace's method is awkward when the angle at the 

 comet in gne of the three observations is very nearly a right one : 

 and Newton's computation cannot be employed, when either 

 the inclination of the orbit is very small, or the earth is very near 

 the line of the nodes in one of the observations. Instead of the 

 hypotheses respecting the time and distance of the perihelium, or 

 the situation of the plane of the orbit, we may make three sup- 

 positions respecting the curtate distances of the comet from the 

 sun, in two of the remotest observations in our possession ; we 

 may compute these curtate distances from the approximate orbit, 

 calling then A' and a'" and then assume 



1 Hyp. 2 Hyp. 3 Hyp. 



1 Obs. a' a' + m a' 



3 Obs. A " A"' a'" + n 



We then compute, for a' and a + wi and the observed position 

 of the comet, the heliocentric longitude and latitude at the time 

 of the first observation, and for a'" and A"' + "the heliocentric 

 longitude and latitude in the third observation. These compu- 

 tations are very easy : for the angle at the projected place of the 

 comet, in the plane of the ecliptic, is found by the equation 



sin c = ^'" ^ ~ "" . observing, however, to take the angle 

 A 



acute or obtuse, according to circumstances ; and hence we 



find e, the elongation of the comet from the earth = 180° 



— c — (A — a) ; and for the heliocentric latitude, tang ^ = 



tang g sin (A — «) . ^^^^ ^^^^^ compute, according to §. 42, the 



sin E 

 longitude of the ascending node, and the inclination of the orbit, 

 for each of the three hypotheses; and since r' = A' sec V and 

 r" =: a'" sec x'", we hence find the true anomalies in both ob- 



