150 Mr. Ivory on cl new Method of deducing 
We must now have recourse to the properties of the orbit 
to get such an equation between the functions, 1 ^, and V as 
shall serve to determine the unknown quantity which they 
contain. For this purpose we may employ the expression 
which gives the time of describing an arc of a parabola, by 
means of the chord of that arc and the two radii vectores drawn 
to its extremities. This elegant property of the motion in a 
parabola, seems to have been first found out by Euler, as 
M. Gauss informs us ; but it is commonly attributed to M. 
Lambert of Berlin, who probably discovered it without know- 
ing what Euler had done, and who has extended it to all the 
conic sections.* It would be superfluous to give here the in- 
vestigation of a truth so well known : it will be sufficient to 
refer to the Mecanique Celeste of Laplace,*!' or to the work of 
M. Gauss. Let h denote the chord drawn between the places 
of the comet at the two extreme observations : then observing 
that T + is the arc of the earth's mean motion correspond- 
ing to the time of describing the parabolic arc of which 6 is 
the chord ; we shall get 
and, by expanding the radicals. 
T 4* = \/r 4- r' . 
1 
48 • (r+r)* 256 * 
and, by squaring and omitting the sixth and higher powers 
of b, 
4 (r-f-r') . 6® . 1 1 
I b* \ 
12 • (r+r ';*3 • 
Let fl® = sr* = (r-J- r*)* 4 — r*Y = 
• Theor. Mot. Corp. Codes. Lib. I. Sect. 3. § 106. 
f Prem. Part. Lib. II. Chap. 4. ^ 27. 
