1 4*^ Mr, Ivory on a new Method of deducing 
is at a considerable distance from the sun, and has a slow mo- 
tion, allowing an interval of 20, or even a greater number of 
days, between the extreme observations ; in which time the 
apparent deviation from a great circle of the heavens will be 
considerable enough to define the magnitude of f with tole- 
rable certainty. But in many circumstances the same mode 
of solution cannot be applied with advantage or success: as 
when a comet has a quick motion, which will allow only a very 
short interval between the observations ; or when a comet 
and the earth are both at the same distance from the sun, or 
nearly so, in which case ^ is evanescent, or very small. 
It appears then, that, generally speaking, the problem of 
the comets must, in reality, be considered as indeterminate, if 
we set aside the condition derived from the nature of the pa- 
rabola. In this respect there is an essential distinction between 
the investigation of the orbit of a planet and that of a comet ; 
a distinction which, when due regard is had to practical utility, 
renders it necessary to separate the particular case from the 
general problem. When the question is to determine a pla- 
net^s orbit by means of three observations made at short in- 
tervals of time, without any assistance from the supposition 
of a circular orbit, or other hypothesis, the equations which 
the data furnish are barely sufficient for finding three radii 
vectores of the orbit and the two angles contained between 
them ; and there are no supernumerary conditions to compen- 
sate what may be imperfect in the observations. In whatever 
manner w^e proceed, the solution will lead to an equation, such 
as that we have been speaking of ; and the success of the inves- 
tigation will therefore depend upon our being able to ascertain 
with some degree of certainty the magnitude of the coefficients 
