CHAPTER XXVI. 



AN INTRODUCTION TO THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 



396. Introduction. 



The Theory of Screws in non-Euclidian space is a natural growth from 

 some remarkable researches of Clifford* in further development of the 

 Theory of Riemann, Cayley, Klein and Lindemann. I here give the in 

 vestigation sufficiently far to demonstrate two fundamental principles 

 ( 42 7&amp;gt; 434) which conduct the theory to a definite stage at which it 

 seems convenient to bring this volume to a conclusion. 



I have thought it better to develop from the beginning the non- Euclidian 

 geometry so far as we shall at present require it. It is thus hoped to make 

 it intelligible to readers who have had no previous acquaintance with this 

 subjectf. I give it as I have worked it out for my own instruction*. It is 

 indeed characteristic of this fascinating theory that it may be surveyed from 

 many different points of view. 



397. Preliminary notions. 



Let #!, # 2 , x 3 , x t be four numerical magnitudes of any description. We 

 may regard these as the co-ordinates of an object. Let y lt y 2 , y 3t y 4 be the 

 co-ordinates of another object, then we premise that the two objects will be 

 identical if, and only if 



1 ^2 3 4 



?/l ~ #2 ~ 2/3 ~ 2/4 



All possible objects may be regarded as constituting a content. 



&quot;Preliminary Sketch of Biquaternions,&quot; Proceedings of the London Mathematical Society, 

 Vol. iv. 381395 (1873). See also &quot;On the Theory of Screws in a Space of Constant Positive 

 Curvature,&quot; Mathematical Papers, p. 402 (1876). Clifford s Theory was much extended by 

 the labours of Buchheim and others ; see the Bibliographical notes. 



t We are fortunately now able to refer English readers to a Treatise in which the Theory of 

 non-Euclidian space and allied subjects is presented in a comprehensive manner. Whitehead, 

 Universal Algebra, Cambridge, 1898. 



Trans. Roy. Irish Acad., Vol. xxvm. p. 159 (1881), and Vol. xxix. p. 123 (1887). 

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