n^idt T the combination with 7. Although we might Mop li< i, , 

 the combinations with S a , S D , S v , and / have now ! 

 ill'iriently discussed, it may yet be of interest to cxt.-nd these 

 ii ions. Of course it will then appear, that really no new group-, 

 be produced beyond those already mentioned. 

 For this purpose let u> tir^t investigate the combination of S r 

 id S. This combination will be equivalent to a rotation round 

 axis A n through an angle which is double that between S v and S D , 



- 



.-. through an angle-. As this rotation is not yet included among 

 haracteristic of D n , -- because the angle of rotation corn 

 Hiding to A n is - , the groups 7) and D% will really be diffe- 



it. But D% need not be considered because we found it identical 

 ath D%', so it is once more confirmed, that D l n and D% are 

 illy different from each other. 



Similarly 5 F and 7 are together equivalent to a rotation through 

 round a binary axis perpendicular to S v . As S v passes through 

 binary axis of D n , the resulting binary axis will be perpendicular 

 one of the binary axes characteristic for D n . If n be even, such 

 axis perpendicular to one of the other binary axes will be found 

 ready among those of D n . If n be odd, this would not be the case, 

 icrefore only if n is an odd number, will the groups D v n and 

 be different; but D% being for all values of the same as 

 it is only demonstrated here once more that D" and D n 

 wholly identical for even numbers n, and only different if n is an 

 number. But, as we have seen, for w = odd, D n is identical with 

 . n already deduced. 



Finally we have to consider the last possible combination: that 

 S D and 7. The result of both is a rotation through 180 round a 

 inary axis perpendicular to S D . Now, if n is odd, such binary axes 

 already have been found among those of D n ; if however ;/ be 

 cvoi, it will be a new one with respect to the binary axes of D n . 

 From this it follows that only for n = even, can the deduced groups 

 D% and D* n be different from each other. But if n is an even number, 

 D n will be always the same as D^', so that our result amounts 

 t<i saying that for even n the groups D% and D 1 ^ will yet differ, - 

 just as we have already stated. Since, because for odd n, D is 

 certainly different from 7)^f, it is here once more demonstrated 

 that in all cases D% and D% must be different from each other. 



S 



