90 Mr. L. Fletcher on the Dilatation of 



Returning to the so-called thermic axes, we have seen that 

 their rectangularity and permanency has been generally 

 assumed for all temperatures. But on consideration we must 

 decide that this is an unfair assumption; for just as Neumann 

 was led by his theory of the symmetrical nature of every 

 crystal with respect to three perpendicular planes to conclude 

 that these perpendicular planes would remain fixed in space 

 for all temperatures, so we might in the same way almost 

 conclude the converse of this proposition, and say that if these 

 planes are permanent in position in space for all temperatures, 

 they are planes of symmetry to the molecules and their group- 

 ing. It is true that if these planes only retain their mutual 

 inclination whilst altering their directions in space, we cannot 

 come to the above conclusion; but in that case we should 

 probably say that thermic axes which revolve with the tem- 

 perature do not bear any very intimate relation to the struc- 

 ture of the crystal — that, in fact, they are mere geometrical 

 inventions, and something akin to the axes of that hypothe- 

 tical ellipse which the moon describes round a certain point as 

 focus. 



For simplicity we shall first consider the alteration of a 

 crystal presenting at least one plane of symmetry common to 

 the molecules and their grouping, and this at all temperatures. 



Imagine a circular cylinder cut at any temperature from 

 such a crystal, with its axis perpendicular to the plane of 

 symmetry. From the above, if the cylinder be freely sus- 

 pended in space and its temperature changed, the axis will 

 remain permanent in direction. In general, the planes perpen- 

 dicular to the axis will be translated parallel to it ; but one 

 plane, that passing through the centre of mass of the crystal 

 or the middle point of the axis, will remain permanently fixed 

 in space ; the crystal -lines, however, which lie in this plane 

 will in general change their directions in space and likewise 

 their mutual inclination. To investigate the nature of these 

 changes, we shall take in this plane rectangular axes OX . OZ 

 fixed in space. At any instant determined by the time t, let 

 the crystal-line of unit length coinciding with OX be rotating 



in space with the angular velocity -=~ or #, and be increasing 



in length at the rate — ora; and let -~ or 0, ~ or y, be the 



corresponding values for the crystal-line of unit length coin- 

 ciding at the same instant with the direction OZ. If x z be 

 the coordinates of any point P at this instant, its resolved ve- 

 locities in the directions OX.OZ will be tcu—zcj), xd + zy 

 respectively. If the change of OP at this instant be one of 



