( 4oft ) 



Crystallography. — "On the iimiu'ssih/c on/i'.rs of the axes of 

 syimrietrji in criistaUoiirai>h\ .'' By Prof. W. \'oi(;t al G()j tingen. 

 (Comninnicatcd by H. A. Lorkntz). 



(Comnnmicalcil in llie meeling of November 30, 1907.) 



Ill one of tlie arlicles of llie seeoiul pai'l of liis colleckMl papers, 

 H. A. Lorkntz (ook up the (picslion — wliicli is e(pially iin[)()rtant 

 in belli erys(allogi'a[)liy and crystaljtliysics — of the |)eniiissil)le 

 order of an axis of syinnielry of the lirst oi- the second kind. In this 

 investigation lie jH'oeeods froiii the principle of the rational duplicate 

 ratio, from which he liisl proxcs that it is consistent with itself, and 

 therefore a suitable basis for crystallographical dednctions. 



The study of this interesting- treatise led nie to the thought, that 

 for the purpose at hand another fnndainental principle of crystallo- 

 graphy — viz. that of the rational indices — might xvell form a 

 more convenient starting point. The continuation of this thought led 

 me to the following develo[)meut, which, 1 belie\e, attains the end 

 in \ iew in a remarkably simple and short manner. 1 \\\\\ prove for 

 this useful fundamental priiici[)le, as Loi{kntz did for the principle 

 of the rational du|)licate ratio, that it does not contradict itself and 

 then derive from it the permissible orders of the axes of symmetry. 



1. The i)riiiciple of the rational indices, as is well known, is as 



follows. 



If we select three arbitrary boundary surfaces of a crystal pol^v- 

 hedroii and draw through any point O j)arallels to their lines of 

 intersection to foi'iii a system of axes ; if we choose further two 

 other arbiti'ary positions thr(»ugh this system of axes, and then the 

 intercepts of these planes upon the axes are 



V. = OA, V = OB, ?" = OC on the one hand, 

 u'=OA\ r'=(>B', ir' = OC' on the other, 

 the principle of the rational indices maintains then, that, 



^ï V 'w' 



-:-■- = z,:^,:~, (1) 



71 r )r 



forms at all limes a ratio of whole numbers. 



In order that this principle should lead to no contradiction, it is 

 necessary that if one ])rocee(ls from three other boundary surfaces of 

 the polyhedron and uses ihe/'r lines of intersection as the fundamental 

 system of axes, then the polyhedral surfaces have also on these rwes 

 intercepts with the above mentioned relation, if the principle held 

 with reference to the lirst system of axes. 



