ig2 Mr, Ivory on a nezv Method oj deducing 
the errors of observation, always considerable of themselves 
in the case of the comets. These remarks, which are cer- 
tainly just, are made by Legendre in his memoir on this pro- 
blem, published in 1806. That geometer therefore thinks it 
preferable to deduce the orbit immediately from the observed 
places of a comet: and w^e agree with his opinion, that the 
observations will be better represented this way, than by em- 
ploying the method of Laplace. With regard to the solutions 
of the problem which Legendre himself has given, it will be 
admitted that they approximate sufficiently near to the ele- 
ments of the orbit, to answer all the purposes of practical as- 
tronomy ; but his formulas are complicated ; and the number 
'of equations which it is necessary to form and to solve, seems 
to render his methods ill adapted for general use. It is the 
object of this paper to give a new solution of this problem, 
which, while it does not yield to any of the known methods 
in accuracy of result, I judge, will be found as commodious in 
practice as the nature of such a calculation can well admit. 
1. Let the coordinates that determine the position of a 
comet with regard to the ecliptic, and a straight line drawn 
through the first point of Aries, (which point is supposed to 
be immoveable) be represented by x,y^%\ of which the last 
is perpendicular to the plane of the ecliptic, and the other two 
have their origin in the sun's centre : and likewise let the 
comet's distance from the sun, or the radius vector of the or- 
bit, be represented by r. Farther, supposing the mean dis- 
tance of the earth's orbit to be unit, put m for the length of 
the circular arc of the mean motion described in a second of 
time; then ~, the versed-sine of that arc, will be the space 
