ROTATIONS IN SPACE OF EVEN DIMENSIONS. 173 



distributive since the operations in terms of which it is expressed are 

 distributive. Equation (26) shows that <f>i and 4>o are commutative if 



iVl * J/2= 0. 



Conversely, if 0i and 02 are commutative and none of the angles 

 equal to t. 



il/l*il/2=0. 



For we can write (f)i in the form 



ct>i = cm I + «i2 (/-il/i) +....+ ai.„ (/-^/i)-! 

 as in (20). If then 02 is commutative with 0i 



4>2 01 - 01 02 = fll2[02 (I -Ml) - (Z- J/l)02] +....+ fll^[02(/- J/l)'"-^ 



- (/•3/)-l02] = 0. 



Since 02 will also be commutative with 0i-, from (21) 

 a22 [02 (I- Ml) - (/• 3/i) 02 + .... + a2402(/- Ml)-! 



- (/•il/i)'"-l02] = 0. 



Similarly for other powers of 0i. Finally 



Omm-i WI-M,) - (/•3/i)02] + . . . + a,„_i„[02(Z-3/i)'"-i 



- (/•Mi)-102] =0. 



There are m — 1 equations linear and homogeneous in m — 1 un- 

 knowTis [02(/-J/i) — {I -Ml) 02], etc. Since the determinant (23) 

 is not zero each of the unknowTis must be zero. Hence 



02(/-il/i) - (7 -3/1)02 = 0. (28) 



This shows that 02 is commutative with I- Mi. Expanding 02 in the 

 form (20) and repeating the argument, we then get, 



(I -Ml) il-M,) - (I-M.) (I- Ml) = 0, 



