a first Approximation to the Orbit of a Comet, 1 45 
before observed, when the comet's orbit coincides with the 
ecliptic, in which case sin. z = 0; and likewise when the great 
circle drawn through the two extreme geocentric places of the 
comet cuts the ecliptic in the places of the earth and the sun 
at the middle observation, in which case sin. — zz) = 0. 
The inferences here drawn from the preceding analysis 
coincide with the rules first given by M. Lambert, of Berlin, 
for judging of a comet's distance from the sun by the inflection 
of its apparent path. 
6 . The second of the equations (10) contains only one un- 
known quantity, namely p° : and hence it may be thought that 
we have already, by means of that equation, obtained a solu- 
tion of the problem, which is both simple and elegant. And 
this w’ould undoubtedly be the case, were it not that the co- 
efficient ^is always extremely small and greatly affected with 
the errors of observation. It depends entirely on the deviation 
of the comet's apparent path from a great circle of the heavens; 
and this deviation is often so little, that small changes in the 
observed places of the comet, by no means inconsistent with 
the errors of observation, will make ^ evanescent, or even 
take a different sign from what it had before. If we suppose 
the motions both of the earth and the comet to be rectilineal 
and uniform, which is never far from the truth in the short 
intervals that must intervene between the observations selected 
for finding a comet's orbit; then the apparent motion of the 
comet w^ould be accurately in a great circle of the heavens, 
and the mode of solution here alluded to could not be applied 
at all. 
The equation here spoken of may no doubt be usefully ap- 
plied in favourable circumstances, particularly when the comet 
MDCCCXIV. U 
