20 Mr. W. Barlow on Crystal Symmetry. 



same, but presents the same orientation, always form some 

 parallelepipedal network of points. It was by postulating 

 such networks that Haiiy deduced the law of rational indices, 

 the experimental truth of which he afterwards established by 

 observation ; and his work shows that if any three homogene- 

 ous lines of points of any such network, which intersect in a 

 single point, be chosen as axes, the intercepts on each of these 

 axes made by the homogeneous planes of points o£ the net- 

 work bear rational relations *. 



The Possible Variety of Relative Arrangement of Directions 

 found Identically related to Ultimate Structure. 



Tt has been stated above f that homogeneous structures, 

 when their directional symmetry alone is considered, are of the 

 si me class when the number and arrangement of like direc- 

 tions is the same in them, and it has been shown that for 

 every homogeneous structure in which two or more directions 

 are identical, a sphere of reference can be constructed, with an 

 axis or axes through its centre, so that the repetition of identical 

 linear or plane directions in the sphere corresponds exactly to 

 that prevailing in the homogeneous structure J. 



It is therefore immaterial to the present inquiry to ascer- 

 tain what, in any given instance, the relative arrangement of 

 parallel axes may be ; all the axes in any given direction will 

 be represented on the sphere of reference by a single axis 

 which will have the direction of the axes which it represents 

 and display coincidence-rotations capable of producing all 

 the different changes of orientation effected about these 

 axes. 



Both in the homogeneous structure and in its sphere of 

 reference, the minimum coincidence-rotation about the given 

 direction will involve the occurrence of all the changes of 

 orientation taking place about this direction : in other words, 

 this rotation will produce a change of orientation which is an 

 aliquot part of every one of these changes of orientation. 

 This becomes evident when it is considered that the existence 

 of a minimum coincidence-rotation a involves that of rotations 

 2 a, 3 a, &c§, and that no other rotation than one of these is 

 possible, since, if such existed, a rotation whose amount is the 

 difference between that of the latter and the nearest of the 

 values referred to, p x, would also be possible, i. e. some less 



* Hatty's Traite de Mineraloyie, 1831, torn. i. p. 283. 

 t See pag-e 6. \ See page 11. § See page 13. 



