536 THE THEORY OF SCREWS. 



Dynamics the author introduces the complex numbers called Biquaternions by 

 Clifford. Again, Mr Chawner translates : 



&quot;The more 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 have 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 

 a + wb where w is a symbol with the property o&amp;gt; 2 and at once all formulae in the 

 Theory of Quaternions could be regarded as formulae in the Theory of Biquaternions. 

 Second, I found that to the various operations in biquaternions 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 

 Theory of Screws. This I call the Theory of Transference and devote a great 

 part of my book to it.&quot; (Theory of Vectors Kasan, 1899.) 



He also mentions the Screw Integrals of certain differential equations and says, 

 &quot;If from two screw integrals corresponding to two given screws (we will call them 

 a and /?) we construct a third with the aid of Poissori s brackets, then the screw of 

 the latter will be the vector product of the screws a and /3 of the given integrals. 

 This circumstance allows us to use biquaternions in order to investigate the 

 properties of screw integrals and their groups.&quot; 



BALL (R. S.) The Twelfth and concluding Memoir on the Theory q/ 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. 

 I could not develop the complete theory until I had obtained a geometrical method 

 for finding the instantaneous screw from the impulsive screw. This has been set 

 forth in this Memoir, and in Chap. xxn. of this volume. 



RENE DE SAUSSURE. Principle* 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 / is a geometrical unit 

 symbol like V- 1. The quantity (P+QI) + f will be regarded 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 

 of the composition of wrenches are obtained. 



M c AuLAY (Alex.) Octonions, a development of Clifford s Bi-quaternions. 8vo., 

 pp. 1-253. Cambridge (1898). 



&quot; An octonion is 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 other its convert. The axis of the motor is called the axis of the octonion.&quot; 

 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 Poirisot s theory of rota 

 tion is also given by the octonion methods. On p. 250 Mr M c Aulay has kindly 

 pointed out that the &quot;reduced wrench&quot; is a conception which cannot have place in 



