Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/31

 If the variations $$\delta \omega_{1}, \delta \omega_{2}$$, etc. are capable both of positive and of negative values, we must have  To illustrate the use of these equations in a case in which $$d\omega_{1}, d\omega_{2}$$, etc. are not exact differentials, we may apply them to the problem of the rotation of a rigid body of which one point is fixed. If $$d\omega_{1}, d\omega_{2}, d\omega_{3}$$ denote infinitesimal rotations about the principal axes which pass through the fixed point, $$\Omega_{1}, \Omega_{2}, \Omega_{3}$$, will denote the moments of the impressed forces about these axes, and the value of $$U$$ will be given by the formula where $$a, b$$, and $$c$$ are constants, $$a + b, b + c, c + a$$ being the moments of inertia about the three axes. Hence,

and the equations of motion are