ROTATIONS IN SPACE OF EVEN DIMENSIONS. 179 



Since both sides are distributive functions of xi, ro, tts it is sufficient 

 to prove the identity when they are products of units. If ir2= tts or 

 7r2= TTs both sides vanish. In any other case with a proper choice 

 of axes we could write 



TTo 



= A*12, TTs— ^'13, TTi— I'ij. 



Since the subscripts 2 and 3 enter in a similar manner, we need con- 

 sider only the four cases 7ri= ku, ku, ^'23 or A-24. In these cases the 

 identity becomes 



- kn= - A-13, = 0, = 0, A-43= - A- 



34 



respectively. 



By applying (38) to each term and adding, since 7ri-7r2'= 7r/-7r2 

 ecc, we get 



TTi * f T2 * TTs) + TTo * (tTs * TTi) + TTs * (tTi* TTo) = 0. (39) 



Another identity is 



(tTi * To) * (tTs * 7r4) = (tTi • TTs) T2 * TTi— (tTi • ^4) TTo * TTs 

 -\- (tTo" Ti) TTi* TTs — {t-z' TTs) TTi * TTi 

 -\- (tTi • TTs') TTi * -Ki — (tTi ' 7r4 ) 7r2 * TTs 

 + (7r2 • TTi) TTi * TTs — (^2 • TTs ) TTi * 7r4 . 



(40) 



Both sides vanish when 7ri= tto, 7ri= tto', 7r3= 7r4 or ts= ta. Since 

 there are only three units and their complements, we must have in 

 any other case tti or ir2 equal to its, tts, tta or in'. These cases are 

 all handled in practically the same way. We consider then only the 

 case 7ri= tts. The identity then reduces to 



(tTi * 7r2) * (tTi * 7r4) = (tTi'TTi) 7r2* 7r4. (41) 



Both sides vanish when T2= Ti or X2= ta. We could then take 



7ri= A'i2, 7r2= A'i3, 7r4= A'i4 or A;23' 

 In these cases (41) becomes 



A"23 A"24= A'34 



