If \ bar { h } \, is the perturbing force and \ bar { V } \, is the velocity vector of the Kepler orbit the equation ( ) takes the form:
12.
The corresponding difference between multiples of the mean motions,, corresponds to a period of nearly 900 years, and it occurs as a small divisor in the integration of a very small perturbing force with this same period.
13.
The perturbing force is sometimes in such cases called a tidal force ( for example, the perturbing force on the Moon ) : it is the difference between the force exerted by the third body on the second and the force exerted by the third body on the first.
14.
The perturbing force is sometimes in such cases called a tidal force ( for example, the perturbing force on the Moon ) : it is the difference between the force exerted by the third body on the second and the force exerted by the third body on the first.
15.
After introducing his diagram'to find the force of the Sun to perturb the Moon'in Book 3, Proposition 25, Newton developed a first approximation to the solar perturbing force, showing in further detail how its components vary as the Moon follows its monthly path around the Earth.
16.
Laplace found that the effects of the solar perturbing force, and of the planet s oblateness ( its equatorial bulge ), together gave rise to an " inclinaison propre ", an " own inclination ", in the plane of the satellite orbits, relative to the plane of Jupiter s equator.
17.
Its use in this context derives from the fact that, at any point in time, an object's osculating orbit is precisely tangent to its actual orbit, with the tangent point being the object's location and has the same curvature as the orbit would have in the absence of perturbing forces.
18.
The main perturbing force to be counter-acted to have a frozen orbit is the " J _ 3 \, force ", i . e . the gravitational force caused by an imperfect symmetry north / south of the Earth, and the " classical theory " is based on the closed form expression for this " J _ 3 \, perturbation ".
19.
He also took the first steps in investigating how the perturbing force shows its effects by producing irregularities in the lunar motions . ( In this part of the enterprise, Newton's success was more limited : it is relatively uncomplicated to define the perturbing forces, but heavy complexities soon arise in the problem of working out the resulting motions, and these were to challenge mathematical astronomers for two centuries after Newton's initial definition of the problem and indication of the directions to take in solving it .)
20.
He also took the first steps in investigating how the perturbing force shows its effects by producing irregularities in the lunar motions . ( In this part of the enterprise, Newton's success was more limited : it is relatively uncomplicated to define the perturbing forces, but heavy complexities soon arise in the problem of working out the resulting motions, and these were to challenge mathematical astronomers for two centuries after Newton's initial definition of the problem and indication of the directions to take in solving it .)
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