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since the terms on the right-hand side cancel in pairs. The centre of gravity has position vector rc, defined by Mrc = Y,, m, r,, M being the sum of all the masses, and so it follows that rc = 0. This can be integrated twice to give rc = at + b, and the constant vectors a and b represent six constants of integration.

There are four more integrals that can be derived easily, all of which were known to Euler. If we take the vector cross product of Eqn (9.13) with r, and then sum, we obtain Y,, m, r, x r, = 0, which can be integrated once to give mjr, x r, = h, (9.15)

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