A Treatise on the Theory of Screws

532 THE THEORY OF SCREWS. Chap. ii. is devoted to groups of screws. He discusses in detail groups con- taining three members, investigating some special cases not dwelt on before. He then gives the formulæ for what are termed “Oblique Co-ordinates.” In Chap. in. the application of the Theory of Screws to mechanics is discussed and the leading parts of the Theory of Screws in relation to dynamical problems with freedom of the «th order are set forth, and he adds, “The lack of books on the Theory of Screws both in Russia and abroad makes us hope that our work will be received -with indulgence.” Ball (R. S.)—The Theory of Permanent Screws. Ninth Memoir. Transactions of the Royal Irish Academy, Vol. xxix., pp. 613—652. 1890. Using Screw-chain co-ordinates an emanent (see Salmon’s Higher Algebra, § 125, or Elliott’s Algebra of Quantics) is here shown to vanish. This involves a general property of the function T which expresses the kinetic energy. dT dT „ «1 . + Xn -j— = 0. doc It is shown that for the permanent screw-chains, aaij dxn The special cases for the different degrees of freedom of a single rigid body are considered in detail. If the rigid body has three degrees of freedom then there are three permanent screws, about any one of which the body will continue to twist if once set twisting. In general, if the body be set twisting about a screw 0, a restraining wrench on some other screw y would be necessary if the motion were to continue as a twist about 0. To find y we employ the plane representation. We construct first a system of points homographic with the points 0. The double points of the homography are representative of the three permanent screws. If we draw the ray connecting 0 with its correspondent, then y is the pole of this ray with respect to the conic of zero pitch, while the pole of the same ray with respect to the conic of inertia gives the screw about which the acceleration is imparted to 0. For freedom of the first and second orders there is only one permanent screw ; for freedom of the third, fourth, and fifth, there are three permanent screws. When the body is quite free the permanent screws are triply infinite. The Theory of Permanent Screws is given in Chap. xxv. Henrici (O.)—The Theory of Screws. Nature, xlii. 127-132. London, 1890. Under the form of a review of the work of Gravelius (see p. 531) we have here an original and suggestive discussion of the entire subject. Professor Henrici has pointed out several promising lines along which new departures might be taken in the further development of the present theory. Küpper (C.)—Die Schraubenbewegung, das Nullsystem und der lineare Complex. Monatshefte für Mathematik und Physik. Vienna,. 1890, pp. 95-104. In this the theory of the linear complex has been developed from the Theory of Screws. The object appears to have been to introduce the study of the subject into the High Schools in Germany.