Crystals on Change of Temperature. 9 1 



simple elongation, we must have 



xol — z<j) x9 + zy 



x z ' 



OT ... 



6x 2 -(u-y)xz + (f)z 2 = (1) 



We may first remark that there is either a single pair of 

 such lines, whether real or imaginary, or an infinity of them ; 

 for so long as the coefficients do not all vanish, this equation 

 represents only two lines ; while if all the coefficients do va- 

 nish, and thus the equation reduce to any identity, we must 



have = cf) = O and a = 7, in which case the change is one of 

 simple linear dilatation in all directions. If the crystal pre- 

 sent the common symmetry-planes characteristic of the cubic, 

 tetragonal, or hexagonal systems, it follows from the perma- 

 nency of direction of those planes that there will be more than 

 two lines in certain of these symmetry-planes likewise perma- 

 nent in direction in space : in these cases, therefore, the change 

 of every line of the system lying in the common symmetry- 

 plane containing these lines of fixed direction will be one of 

 simple elongation without rotation. In the case of the ortho- 

 rhombic system, the two lines in a symmetry-plane which 

 remain unchanged in direction in space will be the same for 

 all temperatures, and, moreover, will be at right angles. 



We shall now show that these lines, in the case of an oblique 

 crystal, are real, though . it is clear that in general they are 

 not at right angles. From dynamical considerations it is 

 known that, so long as the forces acting on a material system 

 are internal, and therefore of the nature of actions and reac- 

 tions, there can be no change of the moment of momentum of 

 the system about any line. As the molecular system under 

 consideration starts from rest, the moment of momentum of 

 the system about any line must therefore be zero throughout 

 the motion. Further, the moment of momentum of a material 

 system about any straight line is known to be equal to the 

 moment of momentum of the system, supposed collected at its 

 centre of mass and moving with it, plus the moment of mo- 

 mentum of the system relative to its centre of mass about a 

 straight line drawn parallel to the given straight line through 

 the centre of mass. In the case of a molecular group, the 

 latter of these terms will be either absolute zero, or, at any 

 rate, of an order of magnitude which may safely be neglected 

 in comparison with the former term. If, then, r be the distance 

 of the centre of mass of a molecular group of mass m from 

 the above axis, co the angular velocity at this instant of the 

 radius vector perpendicular to the axis and passing through 



