A Treatise on the Theory of Screws

538 THE THEORY OF SCREWS. (e) Examine the scalar and vector parts of the quaternion </>Å. X-1. Show that the pitch of any wrench of the system is inversely proportional to the square of that radius of a certain quadric which is parallel to its axis; also that the locus of feet of perpendiculars drawn from the origin to the central axes of the system is a surface (Steiner’s Quartic) containing three double lines intersecting in the origin. (/) The screws (/z, Å) and (//, Å') being reciprocal if SpX + Shp = show that the screws reciprocal to the system p = </>X belong to the system p = - or that a linear vector function and the negative of its conjugate determine, respectively, a 1 three-system’ of screws and its reciprocal ‘three-system.’ Two other theorems communicated to me by Professor Joly may also find a place here. If a body receive twists about four screws of a three-system and if the ampli- tude of each twist be proportional to the sine of the solid angle determined by the dii-ections of the axes of the three non-corresponding screws, then the body after the last twist will have regained its original position. If four wrenches equilibrate and if their axes are generators of the same system of a hyperboloid, their pitches must be equal. Whitehead (A. N.)—Universal Algebra, Vol. i., Cambridge (1898), pp. i-xxvi, 1-586. It would be impossible here to describe the scope of this important work, the following parts of which may be specially mentioned in connection with our present subject. Book v. Chap. i. treats of systems of forces, in which the inner multiplication and other methods of Grassmann are employed. Here as in many other writings we find the expression Null lines, and it may be remarked that in the language of the Theory of Screws a null line is a screw of zero pitch. Chap. ii. of the same book contains a valuable discussion on Groups of Systems of Forces. Here we find the great significance of anharinonic ratio in the higher branches of Dynamics well illustrated. Chap. in. on Invariants of Groups continues the same theories and is of much interest in connection with the Theory of Screws. Chap. iv. discusses among other things the transformation of a quadric into itself, and is thus in close connection with Chap. xxvi. of the present volume. Whitehead’s book should be specially consulted in the Theory of Metrics, Book vi. The Theory of Forces in Elliptic Space is given in Book vi. Ch. 3, in Hyperbolic Space in Book vi. Chap. 5, and the Kinematics of Non-Euclidian Space of all three kinds in Book vi. Chap. 6. There are also some passages of importance in Statics in Book vn. Chaps. 1 and 2, Book vm. Chap. 4, and on Kinematics in Book vii. Chap. 2 and Book vm. Chap. 4. The methods of Whitehead enable space of any number of dimensions to be dealt with almost as easily as that of 3 dimensions. Study (E.)—Eine neue Darstellung der Kräfte der Mechanik durch geometrische Figuren. Berichte über die Vei’handlungen der königlich-sächsischen Gesellschaft der Wissenschaften zu Leipzig. Mathematisch-physische Classe, Vol. li., Part ii., pp. 29-67 (1899). This paper is to develop a novel geometrical method of studying the problems referred to.