ROTATIONS IN SPACE OF EVEN DIMENSIONS. 



169 



(19). Squaring both sides of (20) and using (19) as before 



<^-= 021 / + (122(1 -M,) + . . . + CloJI-M,)"'-^ 



= 021 / + a22{I-M2) + . . . + (hJI-Mo)""^. 



(21) 



Continuing this process we have finally 



<^"-i= a„._u/ + am-n(I-M,) + . . 

 = ctm-u I + a^-nil-^h) + • • 



. +a,„_i^(7-J/i)™-i 

 . +a„,_i,„(/-3/o)«-i. 



(22) 



Since cj), (jr, (j)^. . . .cf)"^'^, I are linearly independent the determinant 



an 



^22 



Ol3- 

 023 • 



■dim 



Om_12 flm-13- ■ ■ ■dm-ln 



^0. 



(23) 



From (20), (21), (22) we get 



ai2[/-3/i - I-Mo] +....+ ai„,[(Z- J/i)"-!- (Z-i/o)-!] = q, 

 a22[/' J/i - /• J/2] + . . . . a24(/-i/i)'"-i- (/-J/.)-!] = 0, 



a™_io[7. J/i- /•J/2] +....+ a^_U(/- J/ii'^i- (/•3/2)'^i] =0. 



There are 7?i — 1 homogeneous linear equations in m — 1 unknowns 

 (I-Mi- I-Mi), lil-Mi)^- (/• .¥2)2] etc. and since the determinant 

 does not vanish all the unknowns must be zero. In particular 



Hence 



/•3/1- /•M2= /(J/i- J/2) = 0. 

 Ml- 3/2= 



which was to be showm 



The case where one or more of the angles is equal to ir is exceptional. 

 The argument just given does not apply because the characteristic 

 equation of I-M has the quadratic factor 



{I-MY+irU 



corresponding to the linear factor 



