Mr. W. Barlow on Crystal Symmetry. 





Proposition 5. — If in the group of coincidence-movements 

 of a homogeneous structure a screw-spiral movement or a 

 rotation is found, the existence of the axis of this movement 

 involves the existence, evenly distributed in some manner 

 throughout the mass, of axes identical ivith this axis. 



This is obviously a direct consequence of the postulated 

 homogeneity. 



Proposition 6. — Among the identical axes of the last 

 proposition are comprised either parallel axes or such as have 

 their directions inclined to one another at infinitesimal angles 

 {the latter are however presently, by Prop. 9, shown to be 

 impossible) . 



Proof. The different directions taken by the identical 

 axes are either finite or infinite in number. If the number is 

 finite, some out of the infinitude of similar axes found in the 

 infinitely extended structure, must take the same direction : 

 i. e., be parallel. If, on the other hand, the number is 

 infinite, some directions must be but infinitesimally removed 

 from others ; i. e., they must be mutually inclined at infini- 

 tesimal angles. 



Proposition 7. — Two identical axes which are either 

 parallel, or whose directions are inclined at an infinitesimal 



Fig*. 5. 



angle {see Prop. 6) cannot be at a distance from one another 

 which is infinitesimal as compared with molecular dimensions, 

 and this, of course, involves that they cannot cut one another. 

 Proof. Let A, A' (fig. 5) be the points of intersection of 

 the two axes with a plane drawn perpendicular to one of them 

 through the centre P of some space-unit, and let 6 be the rota- 

 tion component of the minimum coincidence-movements about 

 these axes. Then, if the coincidence-movements about A, A! 

 are carried out so that the point P is carried in the one case 

 to P„ in the other to P 2 , these points P x P 2 , which each 

 locate a space-unit of the system, will, in the case of parallel 

 axes, lie precisely, in the case of slightly inclined axes, very 



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