346 



Prof. Miller on the Application of 



the equation of conditiou that a face may belong to a zone, and 

 has sliown that the faces of a crystal may be referred to the axes 

 of any three zones as crystallographic axes, with the aid of ele- 

 mentary geometry only [Nuovo Cimento, vol. iv.). The following 

 investigation shows, that not merely the propositions established 

 by Professor Sella, in which it has not been considered requi- 

 site to adhere closely to the steps of his demonstrations, but all 

 the more important geometrical properties of crystals admit of 

 being easily and concisely proved by the methods of ordinary 

 elementary geometry. The relation between the segments formed 

 by the mutual intersection of four straight lines is frequently 

 used. Though well known, a proof of it is given in order to save 

 the trouble of reference. 



2. Let A, B, C, D, E, F be the points of intersection of four 

 straight lines, as shown in the annexed figures. Let AH, parallel 

 to EC, meet DF in H. Then 



AF . BD = FB . AH, and CE . AH = DC .E A. 

 Therefore AF . BD . CE=FB . DC . EA. 



Fig. 1. 



Fig. 2. 



3. Let XOX', YOY', ZOZ' be any three straight lines given in 

 position, all passing through a given point ; a,h, c any three 

 lines given in magnitude ; h, k, I any three positive or negative 



whole numbers, one or two of which maybe zero. Take 0H=-, 

 measured along OX or OX', according as h is positive or nega- 

 tive; OK = Y" along OY or OY', according as k is positive or 



c 

 negative ; OL = j along OZ or OZ', according as I is positive or 



negative. Let the symbol hkl denote any plane parallel to HKL, 

 including HKL itself, on the same side of the point 0. It is 

 now proposed to investigate some of the properties of the system 

 of planes obtained by giving to h, k, I different numerical values. 



4. Let the point be called the origin of the system of planes; 



