A Treatise on the Theory of Screws

 SsraSfeäÖ 279,280] DEVELOPMENTS OF THE DYNAMICAL THEORY. 297 which is the required expression for the kinetic energy. It is remarkable that the co-ordinates of the rigid body are introduced by the medium of the impulsive screw alone. 280. Expression for the Twist Velocity. If an impulsive wrench of unit intensity on a screw 17 be applied to a quiescent rigid body of unit mass which in consequence commences to twist about an instantaneous screw a, it is required to find the initial twist velocity d. The impulsive wrench may be replaced by component impulsive forces -rjit-.-Vs on the six principal screws of inertia and component impulsive couples with moments —ai)2, br/.,, — byit cr)5, —cy6 about those screws of inertia. The force % is expressed by the velocity it produces in the unit mass parallel to the direction of ,7. The component twist velocity of a is do^ about the first principal screw, and accordingly the velocity of translation parallel to that screw is ada^ Hence we have ad«, = y,, but aa, = —Vi, cos (a?;) whence we obtain* cos (ap) a =--------. Pa It will be noted that in this expression the co-ordinates of the rigid body are introduced through the medium of the impulsive screw alone. A special case arises when the impulsive wrench is a couple, in which case of course pv = 00 . As the effect of an impulsive couple is to produce a pure rotation only, we must under these circumstances have pa = 0. Poinsot’s well-known construction exhibits the axis of the initial rotation as the diameter of the momenta! ellipsoid conjugate to the plane of the impulsive couple. As three conjugate diameters of an ellipsoid could not lie in the same plane, it follows that in the case of pri = 00 we can never have a and 77 at right angles. As p„ is zero while cos (ot^) is not zero, we must have d infinite. This might have been inferred from the fact that as the intensity of the impulsive wrench was not zero while the pitch of the screw on which it lay was infinite, the moment of the impulsive couple was infinite and conse- quently the initial twist velocity must be infinite. * Trans. Roy. Irish Acad., Vol. xxxi. p. 100 (1896).