A Treatise on the Theory of Screws

CHAPTER XVIII . FREEDOM OF THE SIXTH ORDER. 237. Introduction. When a rigid body has freedom of the sixth order, it is perfectly free. The screw system of the sixth order includes every screw in space. The statement that there is no reciprocal screw to such a system is merely a different way of asserting the obvious proposition that when a body is perfectly free it cannot remain in equilibrium, if the forces which act upon it have a resultant. 238. Impulsive Screws. Let A, A, ••• denote a series of instantaneous screws which correspond respectively to the impulsive screws 2?1; R2, ... the body being perfectly free. Corresponding to each pair R1 is a certain specific parameter. This parameter may be conveniently defined to be the twist velocity pro- duced about J-j by an impulsive wrench on Rlt of which the intensity is one unit. If six pairs, j41( Rp, A2, R2, ... be known, and also the corresponding specific parameters, then the impulsive wrench on any other screw R can be resolved into six impulsive wrenches on R1,...R6, these will produce six known twist velocities on Alt... A6, which being compounded determine the screw A, the twist velocity about A, and therefore the specific para- meter of R and A. We thus see that it is only necessary to be given six corresponding pairs, and their specific parameters, in order to determine completely the effect of any other impulsive wrench. If seven pairs of corresponding instantaneous and impulsive screws be given, then the relation between every other pair is absolutely determined. It appears from §28 that appropriate twist velocities about A,,... A? can neutralise. When this is the case, the corresponding impulsive wrenches on Rlt ... R7, must equilibrate, and therefore the relative values of the intensities are known. It follows that the specific parameter of each pair J.!, _Rj is proportional to the quotient obtained by dividing the sexiant of