472 



THE THEORY OF SCREWS. 



[425- 



with, as before, the condition, 



a 2 + /3 2 + r + S 2 = 1- 



If 6 be the intervene through which the vector displaces an object then 

 it is easily shown that cos 6 = c/. 



426. Parallel Vectors. 



The several objects of a content are displaced by the same vector along 

 ranges which are said to be parallel. 



Taking the space representation, 413, Clifford showed that all right 

 vectors, which are parallel, intersect two generators of one system on the 

 infinite quadric, while all left vectors, which are parallel, intersect two 

 generators of the other system. 



A generator intersected by two rays from a right vector may be defined by 

 the points whose coordinates are 



+ &amp;lt; -/3, -7, -B, 



+ & +a , -8, +7, 



while a generator intersected by two rays from a left vector will be 

 defined by 



+ &amp;lt;*o, - fio, ~7o&amp;gt; -&o, 



To prove the theorem, it is only necessary to show that these four points 

 are coplanar, for then the two generators intersect, i.e. are of opposite 

 systems. We have, then, only to show that the following determinant 

 vanishes : 



7 8 



a 



a 



70 



This will be most readily shown by squaring, for with an obvious notation 

 it then reduces to the simple form 











O a ] 



[ /?] 











whence we see that the original determinant is simply 



[ ] [/8o ] 



