aa fl - Q - 



427] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 473 



which expanded, becomes 



(a + ftj3 + 770 + 88 V 

 (a po py 760 + 070 



= a 2 2 + # 2 /3 2 -I- a 2 /3 2 + /2 /3 2 - 7 2 7 2 - S 2 S 2 - &y&amp;lt;? - 7 2 S 2 

 = a 2 (a 2 + /3 2 ) + ^ (a /2 + /3 2 ) - 7 2 (7 2 + S 2 ) - 3 2 (7 2 + S 2 ) 

 = (a 2 -I- /3 2 ) (a,, 2 + /3 2 ) - (7 2 + S 2 ) (7o 2 + 8 2 ) 5 



but, a 2 +/3 2 + 7 2 + 8 2 = 0; 



whence this expression is 



( a /2 + /3 2 )(a 2 +/3 2 + 7 2 +S 2 ) = 0. 



On the supposition that the vectors were homonymous, i.e. both right or 

 both left, the corresponding determinant would have been 



a yS 7 8 



n i ^ 



p a 6 7 



o - @o 7o &o 



/3o a/ -^o 7o 

 Squaring, we get, as before, 



but now, 



whence the determinant reduces to 



[aV] 2 +[/3 ?, 

 a value very different from that in the former case. 



427. The Composition of Vectors. 



Let an object x be conveyed to y by the operation of a vector, and let the 

 object y be then conveyed to z by the operation of a second vector, which we 

 shall first suppose to be homonymous (i.e. both right or both left) with 

 the preceding. Then we have, from the first, supposed right 



yi = + #! + /&r 2 + 73.3 + 8x 4 , 



2/4 = - Bx l 



