34 Mr. W. Barlow on Crystal Symmetry. 



well as the appropriate axes, it is evident that since every 

 parallelepipedal network of mere points has centres of in- 

 version at all of these points, the structure formed with the 

 bodies referred to will, taken as a whole, possess centres of 

 inversion at all the points of the network. The employment 

 of any one of these centres of inversion will locate the set of 

 enantiomorphously-similar directions corresponding to the 

 set of identically related directions previously located, thus 

 performing exactly the same function as a centre of inversion 

 in a centred type*. 



Finally, as to centred types displaying a mirror-image 

 similarity of parts, but which do not possess a centre of 

 inversion : — 



It is evident from the foregoing that every system thus 

 characterized contains half the repetitions of some one of the 

 types above enumerated which have centres of inversion. 

 And also that half of its similar directions and parts, i. e. one 

 fourth of those of the corresponding type possessed of a centre 

 of inversion, bear a mirror-image resemblance to the remain- 

 ing half. Further, that any system of identical directions 

 found in it, the number of which, as just stated, is one fourth 

 the total number of similar repetitions in the type possessed 

 of a centre of symmetry, is of one of the eleven types which 

 are not identical with their own mirror-image. 

 To obtain the types in question : — 



For each of the eleven inversion types of the foregoing 

 table, ascertain which of the eleven types of merely identical 

 symmetry has just one fourth of its repetitions ; the result is 

 shown in the 2nd and 3rd columns of the following table, 

 from which it is seen that in some cases none of the latter 

 are found to fulfil this condition, in others two, in most a 

 single type. 



Rejecting the set of repetitions in the inversion type (the 

 quarter of the points) -which are the inverts of those of the 

 system of lower symmetry just arrived at, ascertain which 

 other quarter of the repetitions of the system of higher 

 symmetry can be added to those of the simpler system to 

 produce a system identical with its own mirror-image, and 

 which the addition of a centre of inversion will convert to 

 the inversion type selected. 

 To take an example : — 



The only quarter system corresponding to that of fig. 

 XVI. is given by fig. III. 



Rejecting the additional repetitions which would convert 



* See p. 32. 



