A Treatise on the Theory of Screws

536 THE THEORY OF SCREWS. Dynamics the author introduces the complex numbers called Biquaternions by Clifford. Again, Mr Chawner translates : The moie I studied these numbers the more clearly I grasped two properties in them to which I assign very great importance. First I found that I had only to nave recourse to a little artifice to make the Theory of Biquaternions perfectly analogous, nay, perfectly identical, with the Theory of Quaternions. I found that I had only to introduce the idea of the functions of complex numbers of the form “ + 0,6 where " is a symbol with the property o? = 0 and at once all formulæ in the Theory of Quaternions could be regarded as formulæ in the Theory of Biquaternions. Second, I found that to the various opei'ations in biquateriiions there correspond various, more or less valuable, constructions of the Theory of Screws, and conversely that to the constructions of the Theory of Screws, which are so important to us, there correspond various operations with biquaternions. To these results I attach great importance. Thanks to biquaternions I can produce perfect parallelism between the constructions and theorems of the Theory of Vectors and those of the Iheory of Screws. This I call the Theory of Transference and devote a great part of my book to it.” (Theory of Vectors Kasan, 1899.) «•n??e a^S° ment'onK/^e Screw Integrals of certain differential equations and says, If ri om two screw integrals corresponding to two given screws (we will call them “ and ß) we construct a third with the aid of Poisson’s brackets, then the screw of the latter will be the vector product of the screws a and ß of the given integrals. This circumstance allows us to use biquateriiions in order to investigate the properties of screw integrals and their groups.” Ball (R. b.) Ike Twelfth and concluding Memoir on the Theory of Screws, with a Summary of the Twelve Memoirs. Twelfth Memoir. Transactions of the Royal Irish Academy, Vol. xxxi., pp. 145-196 (1897). At last I succeeded in accomplishing what I had attempted from the first, could not develop the complete theory until I had obtained a geometrical method tor finding the instantaneous screw from the impulsive screw. This has been set forth m this Memoir, and in Chap. xxn. of this volume. René de Saussure.—Principles of a new Line Geometry. Catholic University Bulletin, Jan. 1897, Vol. iii. No. 1. Washington, D.C. The distance and the angle between two rays are here represented as a single complex quantity known as the Distangle, P + QI, where I is a geometrical unit symbol like V- 1. The quantity (P+Q 1)^-1 wiH fee regarfled as the angular measure of the same interval and will be known as the codistangle formed by the two lines. A Codistangle is a complete representation of a wrench, and the laws ot the composition of wrenches are obtained. M°Aulay (Alex.)—Octonions, a development of Clifford’s Bi-quaternions. 8vo„ pp. l-2o3. Cambridge (1898), A? °ctonion’s a Quantity which requires for its specification and is completely specified by a motor and two scalars of which one is called its ordinary scalar and the othor its convert. The axis of the motor is called the axis of the octonion.” In Chap. v. a large number of examples are given of the applications of Octonions to the Theory of Screws. Many of the well-known theorems in the subject are presented in an interesting manner. A discussion of Poinsot’s theory of rota- tion is also given by the octonion methods. On p. 250 Mr M'Aulay has kindly pointed out that the “reduced wrench” is a conception which cannot have place in