86 Mr. L. Fletcher on the Dilatation of 



the crystallographic systems. It has been shown above that 

 the indices of planes remain constant, and that for this reason 

 the change of position of any plane of the system of given in- 

 dices will be known if we are given the changes of the elements 

 |, i), £ ; a :b : c. Now it may be proved, by help of the ratio- 

 nality of the anharmonic ratios of any four of its planes, that 

 any crystalloid system must present one or other of six types 

 of symmetry, according to the values of £, 77, f and the ratio- 

 nality or irrationality of the ratios a:b : c. For instance, if 



TT a O G 



£ = 77 = J = - andy? -) - be all irrational, the crystalloid system 



of planes will present symmetry to three and only three per- 

 pendicular planes ; while if one and only one of these ratios, 



say j, is irrational and the other elements are unaltered, the 



crystalloid system will present symmetry to the above three 

 planes and also to two additional planes perpendicular to each 

 other and bisecting the angles between two of the former. It 

 becomes an interesting question as to whether or not, in the 

 case of a crystal presenting symmetry to only three perpendi- 

 cular planes, j can ever be so altered by change of tempera- 

 ture as to become rational. In the above memoir Grailich 

 and Lang answer this question in the negative ; for, they argue, 

 if such an event were possible, a crystalloid system of planes 

 might pass, on change of temperature, from rhombic to tetra- 

 gonal symmetry: and this they think is disproved by the state- 

 ment that no crystal has been observed to pass on change of 

 temperature from one type of symmetry to the other, as shown 

 by the fact that the optical properties of a crystal at various 

 temperatures point to a permanent type of symmetry of the 

 luminiferous aether, and therefore of the mass of the crystal. 

 In the first place, we may remark that the difference between 

 an irrational number and the nearest rational one is not very 

 large, being somewhere near the infinitieth decimal figure : if, 



then, 7 experience any finite change, it must necessarily pass 



through a large number of rational values, for every one of 

 which the crystalloid system of planes must possess tetragonal 

 symmetry. The fallacy of the argument is this : it assumes 

 that the symmetry of the crystal, as shown in all its physical 

 properties, is the same as the symmetry of disposition of the 

 crystalloid planes. But it is easy to see that this need not be 

 always the case. 



Suppose the centres of mass of the molecules to be arranged 



