84 Mr. L. Fletcher on the Dilatation of 



will remain parallelograms ; and parallelepipeds will remain 

 parallelepipeds. We may remark, in the first place, that 

 since all the edges of tautozonal planes are parallel, and 

 lines parallel at one temperature are parallel at every other, 

 the property of tautozonality is a permanent one ; and, further, 

 since the condition for the tautozonality of planes, whether 

 crystalloid or not, depends only on their indices, the latter, 

 whether rational or irrational, must also be unaltered by change 

 of temperature. This may also be proved as follows : — Imagine 

 the crystal freely suspended in space : the forces due to change 

 of temperature being all internal and in the nature of actions 

 and reactions, the centre of mass must remain unaltered in po- 

 sition, and may be taken as a fixed origin. At the initial tem- 

 perature let OX.OY.OZ be any three crystal-lines passing 

 through the centre of mass 0, and let ABC . HKL be any two 

 planes cutting them in A, B, C, H, K, L respectively ; if 

 ABC be the parametral plane, the indices of HKL will be 



OA OB OC 

 OH' 0K ; OL' 



At any other temperature let A', B', C, H', K', 17 be the new 

 positions of the crystal-points A, B, C, H, K, L ; the indices 

 of H', K', L'', if A' B' C be the parametral plane, will be^ 



OA 7 OB' PC 

 OH" OK" OL 7 ' 



But points which are in a right line and equidistant from each 

 other at one temperature will possess these properties at any 

 other, and thus parts of a crystal-line which have any ratio 

 at one temperature will have the same ratio permanently ; we 

 therefore must have 



OA' OA 



OB' OB 



OC OC 



OH' OH' 



OK' 0K ? 



OL' OL 



and the indices, whether rational or irrational, are invariable. 

 Since parallelograms remain parallelograms, it follows, from 

 the known properties of nomographic figures, that a circle will 

 in general become an ellipse, and a sphere an ellipsoid. This 

 may, however, be shown very simply as follows: — being, as 

 before, the centre of mass, let OA . OB . OC be three perpen- 

 dicular crystal-lines at the first temperature ; construct a paral- 

 lelepiped having OA . OB . OC for adjacent edges, and let P 

 be the other extremity of the diagonal which passes through 0; 

 at a second temperature the lines OA . OB . OC will in general 

 not only have taken new directions in space, but have varied 

 in mutual inclination and in length. Let OA', OB', OC be 



