A Treatise on the Theory of Screws

352] THE THEORY OF SCREW-CHAINS. 387 %, i) by twists about which those displacements could have been effected. In the construction of an eighth chain, 0, we may proceed as follows:— Choose any arbitrary screw 61. Decompose a twist on into components on «J, ßlt ylt 8lt e-i, This must be possible, because a twist about any screw can be decomposed into twists about six arbitrary screws, for we shall not discuss the special exception when the six screws belong to a system of lower order. The twists on alt ... &c., determine the twists on the screw-chains a, ... %, and therefore the twists on the screws a2, ... which compound into a twist on 62, similarly for ffs, &c.; consequently a screw-chain of which is the first screw, and which belongs to the system, has been constructed. This is, however, only one of a number of screw-chains belonging to the system which have for their first screw. The twist on might have been decomposed on the six screws, ß2, y2, 8l; e1; and then the screws 02, &c., might have been found as before. These will of course not be identical with the corresponding screws found previously. Or if we take the whole seven screws, a1(... T)lt we can decompose a twist on in an infinite number of ways on these seven screws. We may, in fact, choose the amplitude of the twist on any one of the screws of reference, alt for example, arbitrarily, and then the amplitudes on all the rest will be determined. It thus appears that where is given, the screw is not determined in the case of freedom of the seventh order; it is only indicated to be any screw whatever of a singly infinite number. The locus of 3., is therefore a ruled surface; so will be the locus of 3S, &c. and we have, in the first place, to prove that all these ruled surfaces are cylindroids. Take three twists on 01, such that the arithmetic sum of their amplitudes is zero, and which consequently neutralize. Decompose the first of these into twists on «i, /Si, 7i, Si, 77i> the second on a1( ßlt ylt e1( and the third on alt ß2, ylt 8lt e1; It is still open to make another supposition about the twists on let us suppose that they are such as to make the two components on % vanish. It must then follow that the total twists on each of the remaining six screws, viz. alt ßlt ylt 81; elt & shall vanish, for their resultant cannot otherwise be zero. All the amplitudes of the twists about the screw-chains of reference must vanish, and so must also the amplitudes of the resultant twists when compounded. We should have three different screws for 3.2 corresponding to the three different twists on and as the twists on these screws must neutralize, the three screws must be co- cylindroidal. We can, therefore, in constructing a screw-chain of this system, not only choose 3, arbitrarily, but we can then take for 3,2 any screw on a certain cylindroid: this being done, the rest of the screw-chain is fixed, including the intermediate screws. 25—2