94 Mr. L. Fletcher on the Dilatation of 



and by addition of (6) and (7), 



Thus at any instant the sum of the angular velocities of any 

 two perpendicular lines in the plane of symmetry is constant. 

 In the case of an orthorhombic crystal, two lines in a plane of 

 symmetry have been shown to be perpendicular and without 

 angular velocity ; thus at any instant in such a crystal, any 

 two perpendicular lines in a symmetry-plane will be rotating 

 with equal velocities in opposite directions. Again, resolving 



the velocities aot—z<j> f xd + zy along the radius vector OP, we 



get 



*=- ^ = (^cos% + 7sin%)sin% + (acos%-^sin%)cosx 



= a cos 2 x— (<£ — 0) sin x cos X + 7 sin2 X> 

 and similarly, for the perpendicular line, 



tc r = a sin 2 % + (<f> — 0) cos x sin x + 7 cos 2 % 

 and , 



or at any instant the sum of the rates of elongation along per- 

 pendicular lines in the plane of symmetry of a crystal is con- 

 stant for all pairs. 



To sum up, we may say that at any instant the change of 

 configuration of the molecular distribution in a plane of sym- 

 metry may be treated as (1) a simple linear dilatation along 

 two straight lines, in general not at right angles, (2) a linear 

 dilatation along any pair of an infinity of pairs of lines, ac- 

 companied by a rotation of the system as a rigid body : the 

 lines of one of these pairs are at right angles; and the dilata- 

 tion along them is in one case a maximum and in the other a 

 minimum. 



It may be shown in a somewhat similar way that, whatever 

 be the magnitude of the changes, if the crystal-lines of unit 

 length which at a temperature t coincide with the fixed rect- 

 angular axes OX . OZ have become at a second temperature 

 T f of lengths a and 7 respectively, and have revolved through 

 angles of 6 and c/>, the coordinates £ f of the point P' (the 

 position at t' of the point P, of which the coordinates at t 

 were x z), 



% =«<£COS 6— y£sin</>, 



%= ax sin 6 -\-yz cos (p. 



Whence it follows that one pair of lines, whether real or ima- 



