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Solar System, the plane is referred to as the 'invariable plane of Laplace'.

70 Newton Principia, Corollary 4 to the Laws of Motion.

Laplace referred to these integrals as the 'principle of the Chevalier d'Arcy', after the French-Irish mathematician who was one of the first to derive them in analytical form. The appreciation that moments of vector quantities are vector quantities themselves (i.e. they add according to the parallelogram law) can be traced back to the work of Euler who wrote the equivalent of the equation r = w x r for the rotation of a rigid body in 1765 (see Caparrini

See Caparrini (2002). The orientation of this plane is determined largely by Jupiter and Saturn since they contain most of the planetary mass. The ecliptic is inclined at about 11 ° to this plane - which lies between the orbital planes of the two largest planets - though this figure varies over time as the orbit of the Earth is perturbed by the other planets.

The final integral of motion is the energy integral (historically called the 'vis viva integral' or the 'integral of forces viva'). If we dot Eqn (9.13) with r, and then sum over i we obtain v^ v^ • w T, dV

^ mtr, ■ r, = - r, -V,V = - —, i i since V is simply a function of the coordinates of the masses. This can be integrated to give

where T is the kinetic energy defined in Eqn (9.11). The constant E represents the total energy of the system. There are thus ten integrals for the «-body problem that can be shown to be independent, and for the three-body problem they reduce the order of the system of differential equations from 18 to 8.

By changing the independent variable from the time to one of the other dependent variables, Lagrange managed to reduce the three-body problem to a seventh-order system, and he found he could integrate the equations completely if he assumed that the ratios of the distances between the masses were constants. Joseph Alfred Serret, who edited the works of Lagrange (published in fourteen volumes between 1867 and 1892) commented on the 1772 essay on the three-body problem as follows:

The first chapter deserves to be counted among Lagrange's most important works. The differential equations of the three-body problem ... constitute a system of the 12th order, and the solution required 12 integrations. The only known ones were those of the force vive and the three from the principle of areas. Eight remained to be discovered. In reducing the number to seven Lagrange made a considerable contribution to the question, one not surpassed until 1873 ... .

One way to derive Lagrange's exact solutions is to seek solutions for which the resultant force on each body passes through the origin, which is taken to be the centre of gravity of the three bodies. If we cross Eqn (9.2) with r, the above supposition implies that the left-hand side is zero and, hence,

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