The first-order orbital equation

S.

 

Chandrasekhar

, (, Oxford, ), pp. –. In particular, Proposition XLIV-Theorem XIV: The difference of the forces, by which two bodies may be made to move equally, one in a fixed, the other in the same orbit revolving, varies inversely as the cube of their common altitudes.

T.

 

Levi-Civita

, (, Bologna, ), p. .

A. C.

 

Clairaut

, “,” Histoires de l’Academie Royale des Sciences, mem. 1745 and Ref. 4.

We have reproduced many papers of historical interest at $⟨$gallica.bnf.fr/$⟩$⁠.

T.

 

Needham

, (, New York, ).

The Laplace integral can be found in

P. S.

 

Laplace

, (, Paris, ), Tome 1, p. , formula $P$⁠. To obtain Eq. (11), we need to use $z=0$⁠, $c=xẏ−ẋy=ℓ$⁠, $f=μRe(e)$⁠, and $f′=μIm(e)$⁠, and add the first relation to the second one multiplied by $−i$ (see Ref. 4). To keep the customary notation we use the same letter $e$ for the eccentricity and for the complex exponential.

A parallel treatment of the two-body problem with vectorial methods is given by

V. R.

 

Bond

and

M. C.

 

Allman

, (, Princeton, NJ, ).

R. E.

 

Williamson

, (, New York, ), p. .

R.

 

d’Inverno

, (, New York, ), p. .

For a complete calculation of all the perturbing effects see

M. G.

 

Stewart

, “,”  , – ().

Reference 9, p. .

See, for example,

B.

Davies

, “,”  , – ();

N.

Gauthier

, “,”  , – ();

T.

Garavaglia

, “,”  , – ();

C.

Farina

and

M.

Machado

, “,”  , – ();

D.

Stump

, “,”  , – ();

K. T.

McDonald

, “,”  , – ();

S.

Cornbleet

, “,”  , – ();

B.

 

Dean

, “,”  , – ().

R.

 

Adler

,

M.

 

Bazin

, and

M.

 

Shiffer

, , 2nd ed. (, New York, ), p. , Eq. (6.149).

A. C.

 

Clairaut

, “,” Histoires de l’Academie Royale des Sciences, mem. 1745, p. : Clairaut wrote that “The moon without doubt expresses some other law of attraction than the [inverse] square of the distance, but the principal planets do not require any other law. It is therefore easy to respond to this difficulty, and noting that there are an infinite number of laws which give an attraction which differs very sensibly from the law of the squares for small distances, and which deviates so little for the large, that one cannot perceive it by observations. One might regard, for example, the analytic quantity of the distance composed of two terms, one having the square of the distance as its divisor, and the other having the square square.” On p. 362, Clairaut examined the effect of a perturbing inverse-cube force. This memoir is dated 15 November 1747 and can be found in Ref. 4. Clairaut was also the first to introduce a revolving ellipse as a first approximation to the motion of the moon. This idea is sometimes called Clairaut’s device or Clairaut’s trick.

See

F.

 

Tisserand

, (, Paris, ), p. ; reproduced at Ref. 4.

Clairaut was also a first-class geometer, specializing in curvature. See his Recherches sur le courbes a double courbure at Ref. 4.