CERUSSITE CRYSTALS FROM BROKEN HILL AND MULDIVA. 293 



vertical axes, that is the normals to the c pinacoids, have 

 placed themselves parallel in the three segments and in II 

 one zone plane [c w] and normals in that zone are ranged 

 parallel to the corresponding plane and normals in I, while 

 in the r twin III one zone plane [c r] and its normals have 

 placed themselves parallel to the corresponding plane and 

 normals of I, whence the twin relations result. It is 

 apparent that if II and III be rotated slightly cum-clock- 

 wise, the normals to b 2 and Wi r 2 and r l9 m 2 and bi, a 2 and 

 r x and r 2 and d, also r 3 and a u b 3 and Wi, m 3 and bi, a 3 and 

 r l9 m 3 and wii, respectively, will be brought closer together. 

 This will have the effect of bringing b 2 closer to bi and 

 separating b 3 and bi more widely, that is the angle bi A b 2 

 becomes less than the precise 62° 46' and bi a b 3 greater 

 than 57° 18'. In each case therefore the angle b x A becomes 

 nearer to the value 60°. In general none of the axes will 

 coincide but will take a 'middle position.' 



Although the existence of 'planes of force' and 'direc- 

 tions of force ' in the zone planes and face-normals is more 

 or less an assumption, there is no doubt that crystals do 

 possess a directive force by virtue of which the crystal 

 particles, whatever these may be, are marshalled into a 

 regular formation; this is proved by the existence of liquid 

 crystals. Moreover there is nothing unorthodox in speak- 

 ing of parallelism of axes or directions which are not 

 crystallographically equivalent, for we know that crystals 

 do exhibit parallel growths of this kind ; we may cite 

 Goldschmidt's 'hetero- twins,' in which inequivalent but 

 similar and similarly directed axes place themselves 

 parallel or nearly parallel. 1 



Whether Goldschmidt's hypothesis is valid or not it is 

 important to discover whether a similar divergence to that 



1 Goldschmidt, Zeits. Kryst., xliii, 1907, pp. 582-586 ; Goldschmidt 

 and Paul, 16., xlvi, 1909, p. 471; Ford, Amer. Journ. Sci. xxx, 1910, pp. 

 16-23. 



