Rogers — 3fode of Representing Linear Orthogonal Transformation. 71 



Making these substitutions in the linear equation for ?i, »),,?,.... and 

 equating real and imaginary parts, it will be seen after a little linear trans- 

 formation that «„ /3„ 7„ . . . . «2, ^2, 7-3 . • • • satisfy the same equations as 

 the numbers represented by the same letters at the beginning of this section. 

 Hence the flat of rotation corresponding to 0, where* 



cos d = yUi, sin (y = + ^(2 

 is the »S'„.o defined by 



Consider the next angle ^, and proceed as before. Then 



«'■* = /i3 + ifXi, 

 and the corresponding rotatory {n - 2)-flat is defined by 

 Z s 2a3X = 0, IF 3 %aiX = 0. 

 "We also prove that Z and W are orthogonal to X and Y. For 



Hence 2?i?3 = XiXaSljIg. 



But in general A^Aj =[= 1, hence 



S?1?3 = 0, 



which establishes the orthogonality in question. 

 The rotatory fiats are thus defined hy 



X= + P = %U^ = 0, 



Z"- + W^ = 2?3?i = 0, 

 where ^j, ^^, &c., are the ratios satisfying linear equations vjhich give rise to 



/(A) = 0, 

 and the roots of this equation are the various values of e' . 



n - 1 

 When n is odd, the — ^ — rotatory flats are determined in like manner, 



if we take the origin on the " central axis," or axis of uniform translation 

 (§ 22). 



21. In § 20 a special origin was chosen in each case — n even. Here we 



and does not vanish. Hence 



Ml- + /i3= = 1 ; 

 therefore the angle e is always real. 



t On inspection it will be seen that the ambiguity in the sign of e introduces no 

 ambiguity in the transformation. 



