420] THE THEORY OF SCREWS IN NON-EUCLIDIAN SPACE. 465 



In general let X 1 = 0, 



then cos 8 = p3 tjgi 



whence we find that all objects in the extent Z a are displaced through equal 

 intervenes. This intervene can be readily determined for 



whence 



or 



8 = 



The intervene through which every object on X 2 is conveyed has the 

 same value. 



We could have also proved otherwise that objects on X 1 and X^ are all 

 displaced through equal intervenes, for the locus of objects so displaced is 

 a quadric of the form 



XtXt+XX^Tj-t), 



and, of course, for a special value of the distance this quadric becomes 

 simply 



^JT.-O. 



If X l = 0, and X 3 = 0, then cos 8 becomes indeterminate ; but this is as 

 it should be, because all objects on X l and X 3 are at infinity. 



420. The Orthogonal Transformation*. 



The formulae 



y, = (21) Xl + (22) x, + (23) x 3 + (24) x ti 



7/3 = (31) x, + (32) # 2 + (33) x, + (34) a? 4 , 



2/4 = (41 ) x, + (42) x z + (43) ^ + (44) x 4 , 



denote the general type of transformation. The transformation is said to be 

 orthogonal if when x lt &c., are solved in terms of y lt &c. we obtain as follows: _ 



X, = (11) yi + (21) y, + (31) y, + (41) y 4 , 



^ 2 = (12) y, + (22) y a + (32) y, + (42) y 4&amp;gt; 



^3 = (13) yi + (23) y a + (33) y s + (43) y 4 , 



** = (14) y x + (24) y a + (34) y, + (44) y 4 . 

 This is employed in Professor Heath s memoir, cited on p. 452. 

 B &quot; 30 



