Mr. W. Barlow on Crystal Symmetry. 



17 



carrying out the movement proper to A, A' is moved to the 

 place of another similar axis A 7 '. The distance of this third 

 axis from A' cannot be less than A A" since the latter is a 

 minimum distance. Therefore the triangle A A' A" has an 

 angle A which cannot be less than one of the remaining 

 angles ; in other words cannot be less than 60°. Conse- 

 quently n cannot be greater than 6. 



Proposition 13. — Pentagonal axes are impossible. 



Proof. If such axes existed some of them would, by pro- 

 position 6, be parallel. Let A, A' (fig. 7) denote two such. 



Fig. 7. 



Then by carrying out the rotation — — about A a third identical 

 axis A" is located and similarly a fourth A"'. Therefore in 

 the 4-sided figure A' A A" A?", A' A = A A" = A"A'" and 

 L e A' A A"= Z e A A"A'" = 72°; i. e. less than 90°. But A' A'" 

 cannot be less than A A"; therefore ZsA'AA", A A" A"' 

 cannot each be less than 90° and pentagonal axes are im- 

 possible. 



Proposition 14. — Other than digonal, trigonal, tetragonal, 

 or hexagonal axes are impossible. 



Proof. From proposition 11, n must be an integer. 

 From proposition 12 it is not >6, and from proposition 13 

 it cannot be 5. Therefore since the value 1 gives no rotational 

 repetition, it can only be 2, 3, 4, or 6. 



Before proceeding to the consideration of what types of 

 combination can exist of the varieties of axes thus not 

 impossible, reference must be made to an important property 

 of the homogeneous structures above defined which identifies 

 them with the assemblages of sameway-orientated units pos- 

 tulated by Haiiy, Bravais, and others, and generally regarded 

 as necessarily existing in crystals. 



Proposition 15. — Among the coincidence-movements of a 

 homogeneous structure whose ultimate parts are of finite dimen- 

 sions various translations not confined to any one plane are 

 always found, the mass possessing the property that it is 



Phil. Mag. S. 6. Vol. 1. No. I, Jan. 1901. C 



