A Treatise on the Theory of Screws

534 THE THEORY OF SCREWS. given pair. Tn this way the existence of two sets of finite motions, each individual motion of which leaves unchanged each of a doubly-infinite number of straight lines is demonstrated (the right- and left-vectors). It is also shown that, when a right- (left-) vector is represented as the resultant of two rotations through two right angles, the axes of the two rotations are left- (right-) parallels. Hence from the original construction the resultant of two right- (left-) vectors is again a right- (left-) vector. Lastly it is shown, still synthetically, that every right-vector is permutable with every left-vectoi-, thus proving in a different way what is given in § 427 of this volume. See also p. 526. It is also shown that the laws according to which right- (left-) vectors combine together are the same as those by which finite rotations round a fixed point combine. The remainder of § II. is concerned with the determination of all distinct types of “continuous groups” of motions in elliptic space. § III. gives the application of the construction to hyperbolic space. Here attention is first directed to a type of displacement which leaves no finite point or line undisplaced. It is also shown that no displacement in hyperbolic space can leave more than one real line unchanged. This fact, combined with the properties of the previously mentioned special type of displacements, is then used to determine all the distinct types of continuous groups of motion in hyperbolic space. Joly (C. J.)—The Theory of Linear Vector Functions. Transactions of the Royal Irish Academy, Vol. xxx., pp. 597-647 (1894). In this memoir the close connexion between the quaternion theory of linear vector functions and the Theory of Screws is developed. “ The axes of the screws of the resultants of any wrenches acting on three given screws belong therefore to one of the congruencies of lines treated of in the present paper, and every geometrical relation described in it may be applied to problems in Rational Mechanics.” A remarkable quintic surface is discovered which under certain conditions degrades into the cylindroid. At the close of the memoir the linear vector functions expressing screw-systems of the third, fourth and fifth orders are discussed. Ball (R. S.)—The Theory of Pitch Invariants and the Theory of Chiastic llomography. Tenth Memoir. Transactions of the Royal Irish Academy, Vol. xxx., pp. 559-586 (1894). It is shown that if a2 ... a6 be the six co-ordinates of a screw a, while hlt ... hti are the angles which a makes with the six co-reciprocal screws of reference, then expressions of the form ax cos hx + ... + a6 cos are invariants in the sense that they are unaltered for every screw on the same ray as a. If klt ... Z:6 be the similar angles for any other screw, then cos 7t; cos kr cos å2 cos k2 cos h6 cos kf- Pi p2 + ''' 4 ’ where pr ... pt are the pitches of the screws of reference. If two instantaneous screws a and ß and the corresponding impulsive screws V and c are so related that a is reciprocal to f, then ß must be reciprocal to y. lliis clearly implied that there must in all cases be some relation between the virtual coefficients and The relation is here shown to be P* pl3 cos(a>/)®ßl cos^)07“’-