If the deviation be a rotation about an axis traversing the centre of gravity, there is no centre of percussion; for such a deviation can only be produced by a couple of forces, and not by any single force. Let dα be the deviation of angular velocity to be produced in the interval dt, and I the moment of the inertia of the body about an axis through its centre of gravity; then 1⁄2Id(α2) = Iα dα is the variation of the body’s actual energy. Let M be the moment of the unbalanced couple required to produce the deviation; then by equation 57, § 104, the energy exerted by this couple in the interval dt is Mα dt, which, being equated to the variation of energy, gives

M = I dα= R2W· dα.
dt gdt

(83)

R is called the radius of gyration of the body with regard to an axis through its centre of gravity.

Fig. 133.

Now (fig. 133) let the required deviation be a rotation of the body BB about an axis O, not traversing the centre of gravity G, dα being, as before, the deviation of angular velocity to be produced in the interval dt. A rotation with the angular velocity α about an axis O may be considered as compounded of a rotation with the same angular velocity about an axis drawn through G parallel to O and a translation with the velocity α. OG, OG being the perpendicular distance between the two axes. Hence the required deviation may be regarded as compounded of a deviation of translation dv = OG · dα, to produce which there would be required, according to equation (82), a force applied at G perpendicular to the plane OG—

P = W· OG · dα
g dt

(84)

and a deviation dα of rotation about an axis drawn through G parallel to O, to produce which there would be required a couple of the moment M given by equation (83). According to the principles of statics, the resultant of the force P, applied at G perpendicular to the plane OG, and the couple M is a force equal and parallel to P, but applied at a distance GC from G, in the prolongation of the perpendicular OG, whose value is

GC = M / P = R2 / OG.