14 Mr. W. Barlow on Crystal Symmetry. 



approximately in a plane drawn parallel to the plane A A' P, 

 and distant from it /, where I is the translation component of 

 the two movements. 



Let the projections of the points Pj P 2 on to the plane 

 A A' P be P' P". We have then either precisely or very 

 approximately, as the case may be, the two triangles A P P', 

 A' P P" similar, and therefore the triangles A P A', P' P P" 

 also similar. 



Now P P / are two points on the same circle, and therefore 

 the distance P P' cannot be greater than the diameter 2 A P. 

 Therefore A P cannot be infinitesimal as compared with 

 P' P, and so neither can A A 7 be infinitesimal as compared 

 with P' P", and the latter is either equal, or very approxi- 

 mately equal to P x P 2 , which is the distance apart of the 

 centres of some two space-units. 



Proposition 8. — Admitting that the molecular dimensions 

 are finite, it is impossible for any rotation of the structure to 

 have an infinitesimal value, or for any screiv-spiral coincidence 

 movement to have as component an infinitesimal rotation. 



Proof. Among the identical axes of proposition 5 note 

 such as are either parallel or whose directions are inclined to 

 one another at infinitesimal angles, and of these let A A' be 

 two which are at the least distance apart ; this distance must, 

 by prop. 7, be finite. 



By carrying out the movement proper to A, from A' locate 

 a third axis A", which will also form one of the selected 

 flock. Then the distance A' A 1 ' measured in a plane per- 

 pendicular to one of the axes A', cannot be sensibly less than 

 the distance A A', similarly measured, because the axes A A' 

 are at minimum distance apart, and this minimum distance 

 can, exactly in the case of parallel axes, very approximately 

 in the other case, be measured in the plane referred to. 

 Therefore the angle A f A A", which is the angle of rotation 

 of A, must have a sensible value (/. e. not less than 60°, or 

 thereabouts). 



Proposition 9. — An infinitesimal value for the mutual 

 inclination of the directions of two identical axes is impossible. 



Proof. The presence of two identical axes in a structure 

 involves the existence not only of the coincidence-movements 

 proper to these axes, but also the existence of a coincidence- 

 movement, screw-spiral or merely rotational, capable of 

 carrying the structure from an initial position to the ultimate 

 position reached by successively carrying out the two move- 

 ments proper to the two axes *. 



* See page 9. 



