A Treatise on the Theory of Screws

340 THE THEORY OF SCREWS. [314- had been free, would have been the impulsive screws, corresponding to the instantaneous screws belonging to A. Let Q be the system of screw of the (6 — n)th order, which are reciprocal to P. Take from A the system of the (m — l)th order, reciprocal to y, and call it A1. Take from Q the system of the (5 — n)th order, reciprocal to y, and call it Qi. As A is of the nth order, and Q1 of the (5 — n)th, they together define a system of the fifth order. Let be the single screw, reciprocal to this system of the fifth order. As A2 is of the (n— l)th order, and Q is of the (6 —n)th order, they together define a system of the fifth order. Let % be the single screw, reciprocal to this system of the fifth order. Th is reciprocal to of the (n — l)th order, because forms part of A. It is reciprocal to Qi of the (5 — n)th order, because it was so made by con- struction. Thus 7/j is reciprocal to both Aa and that is to a system of the fourth order. In like manner, it can also be shown that is reciprocal to both J., and Qj. and ft were originally chosen so as to be reciprocal to It thus appears that the three screws, r/, 77 b 77,, are all reciprocal to the same system of the fourth order. They are, therefore, co-cylindroidal. The initial wrench on t] can therefore be adequately replaced by two components on 771 and r;.2. The former of these is destroyed by the reaction of the constraints. The latter gives rise to an initial movement on a determined screw of A. Thus, the most general problem of the effect of an impulsive wrench on a constrained rigid body has been solved geo- metrically. 315. Freedom of the Fifth Order. The special case, where the rigid body has freedom of the fifth order, may be viewed as follows. Let p be the screw reciprocal to the screw system of the fifth order, about which the body is free to twist. Let Å. be the instantaneous screw, determined in the way already explained, about which the body, had it been free, would have commenced to twist in consequence of an impulsive wrench on p.