A Treatise on the Theory of Screws

18 the THEORY OF SCREWS. [10- coefficient of the two screws a and ß, and may be denoted by the symbol 11. Symmetry of the Virtual Coefficient. An obvious property of the virtual coefficient is of great importance. If the two screws a and ß be interchanged, the virtual coefficient remains unaltered.. The identity of the laws of composition of twists and wrenches can be deduced from this circumstance*, and also the Theory of Reciprocal Screws which will be developed in Chap. III. 12. Composition of Twists and Wrenches. Suppose three twists about three screws a, ß, y, possess the property that the body after the last twist has the same position which it had before the first: then the amplitudes of the twists, as well as the geometrical rela- tions of the screws, must satisfy certain conditions. The particular nature of these conditions does not concern us at present, although it will be fully developed hereafter. We may at all events conceive the following method of ascertaining these conditions :— Since the three twists neutralize it follows that the total energy ex- pended in making those twists against a wrench, on any screw 7, must be zero, whence a = 0. This equation is one of an indefinite number (of which six can be shown to be independent) obtained by choosing different screws for y. From each group of three equations the amplitudes can be eliminated, and four of the equations thus obtained will involve all the purely geometrical conditions as to direction, situation, and pitch, which must be fulfilled by the screws when three twists can neutralize each other. But now suppose that three wrenches equilibrate on the three screws a, ß, 7. Then the total energy expended in a twist about any screw y against the three wrenches must be zero, whence « + ß"™^ + — 0. An indefinite number of similar equations, one in fact for every screw y, must be also satisfied. By comparing this system of equations with that previously obtained, it is obvious that the geometrical conditions imposed on the screws a, ß, 7, in 4iqT/1i87weSnant re“ark’ °r What is e1uivalent thereto, is due to Klein (Math. Ann., Vol. iv.