A Treatise on the Theory of Screws

257] HOMOGRAPHIC SCREW SYSTEMS. 271 256. Correspondence of m and n systems. We may look at the matter in a more general manner. Consider an »i-system (A) of screws, and an n-system (B) (m>n). (If we make m = 6 and n — 2, this system includes the system we have been just discussing.) To one screw in A will correspond one screw in B, but to one screw in B will correspond, not a single screw in A, but an (m + 1 — n)-system of screws. If m = n, we find that one screw of one system corresponds to one screw of the other system. Thus, if m = n = 2, we have a pair of cylindroids, and one screw on one cylindroid corresponds to one screw on the other. If mi = 3, and n = 2, we see that to each screw on the cylindroid will cor- respond a whole cylindroid of screws belonging to the three-system. For example, if a body have freedom of the second order and a screw be indicated on the cylindroid which defines the freedom, then a whole cylindroid full of screws can always be chosen from any three-system, an impulsive wrench on any one of which will make the body commence to twist about the indicated screw. 257. Screws common to the two systems. The property of the screws common to the two homographic systems will of course require some modification when we are only considering an m-system and an n-system. Let us take the case of a three-system on the one hand, and a six-system, or all the screws in space, on the other hand. To each screw a of the three-system A must correspond, a four-system, B, so that a cone of the screws of this four-system can be drawn through every point in space. It is interesting to note that one screw ß can be found, which, besides belonging to B, belongs also to A. Take any two screws reciprocal to B, and any three screws reciprocal to A, then the single screw ß, which is reciprocal to the five screws thus found, belongs to both A and B. We thus see that to each screw a of A, one corresponding screw in the same system can be determined. The result just arrived at can be similarly shown generally, and thus we find that when every screw in space cor- responds to a screw of an n-system, then each screw of.the n-system will correspond to a (7 — n)-system, and among the screws of this system one can always be found which lies on the original n-system. As a mechanical illustration of this result we may refer to the theorem (§96), that if a rigid body has freedom of the nth order, then, no. matter what be the system of forces which act upon it, we may in general combine the resultant wrench with certain reactions of the constraints, so as to produce a wrench on a screw of the n-system which defines the freedom of the body, and this wrench will be dynamically equivalent to the given system of forces.