A Treatise on the Theory of Screws

318J THE GEOMETRICAL THEORY. 345 the fulfilment of two identical conditions among their coordinates. As there are three pairs of correspondents, we see at once that there are six equations to be fulfilled. These are exactly the number required for the determination of y, %, %, in the system P. To the same conclusion we might have been conducted by a different line of reasoning. It is known that, for the complete specification of a system of the third order, nine co-ordinates are necessary (§ 75). This is the same number as is required for the specification of a rigid body. If, therefore, we are given that P, a system of the third order, is the collection of impulsive screws, corresponding to the instantaneous screws in the system A.t we are given nine data towards the determination of a rigid body, for which A and P would possess the desired relation. It therefore follows that we have nine equations, while the rigid body involves nine unknowns. Thus we are led to expect that the number of bodies, for which the arrangement would be possible, is finite. When such a body is de- termined, then of course the correlation of the screws on the two systems is immediately accomplished. It thus appears that the general problem of correlating the screws on any two given systems of the third order, A and P, into possible pairs of impulsive screws and instantaneous screws, ought not to admit of more than a finite number of solutions. We are now to prove that this finite number of solutions cannot be different from unity. For, let us suppose that a screw X, belonging to A, had two screws 6 and <f>, as possible correspondents in P. This could, of course, in no case be possible with the same rigid body. We shall show that it could not even be possible with two rigid bodies, and M2. For, if two bodies could do what is suggested, then it can be shown that there are a singly infinite number of possible bodies, each of which would afford a different solution of the problem. We could design a rigid body in the following manner Increase the density of every element of in the ratio of : 1, and call the new mass M-[. Increase the density of every element of M2 in the ratio of p2: 1, and call the new mass M2. Let the two bodies, so altered, be conceived bound rigidly together by bonds which are regarded as imponderable. Let be any screw lying on the cylindroid (0, </>), Then the impulsive wrench of intensity, -i//" on xjr, may be decomposed into components on 0 and 011 *• Y sm(0-</>) sin(0 — </>)