A Treatise on the Theory of Screws

Mil [351, 386 THE THEORY OF SCREWS. 351. Freedom of the sixth order. In freedom of the sixth order we select at random six displacements of which the mass-chain admits, and then construct the six corresponding screw- chains. The homologous screws in this case lie on six-systems, but a six- system means of course every conceivable screw. It is easily shown (§ 248) that if to one screw in space corresponds another screw, and conversely, then the homography is completely established when we are given seven screws in one system, and the corresponding seven screws in the other. Any eighth screw in the one system will then have its correspondent in the other imme- diately determined. It is of special importance in the present theory to dwell on the type of homography with which we are here concerned. If on the one hand it seems embarrassing, from the large number of screws concerned, on the other hand we are to recollect that the question is free from the complication of regarding the screws as residing on particular «-systems. Seven screws may be drawn anywhere, and of any pitch; seven other screws may also be chosen anywhere, and of any pitch. If these two groups be made to corre- spond in pairs, then any other screw being given, its corresponding screw will be completely determined. Nor is there in this correspondence any other condition, save the simple one, that to one screw of one system one screw of the other shall correspond linearly. Six screw-chains having been found, a seventh is to be constructed. This being done, the construction of as many screw-chains as may be desired is immediately feasible. From the homographic relations just referred to we have appropriate to each element of the system seven homologous screws, and also appropriate to each consecutive pair of elements we have the seven homologous intermediate screws. An eighth screw, appropriate to any element, may be drawn arbitrarily, and the corresponding screw being con- structed on each of the other systems gives at once another screw-chain about which the system must be free to twist. When a mass-chain has freedom of the sixth order we see that any one element may be twisted about any arbitrary chosen screw, but that the screw about which every other element twists is then determined, and so are also the ratios of the amplitudes of the twists, by the aid of the inter- mediate screws. 352. Freedom of the seventh order. Passing from the case of six degrees of freedom to the case of seven degrees, we have a somewhat remarkable departure from the phenomena shown by the lower degrees of freedom. Give to the mass-chain any seven arbitrary displacements, and construct the seven screw-chains, a, /3, y, 8, e,