Crystals on Change of Temperature. 85 



these new directions and lengths ; on these lines as adjacent 

 edges construct a parallelepiped, and let P' be the other extre- 

 mity of the diagonal which passes through : since parallele- 

 pipeds at one temperature remain so at any other, it follows 

 that P' is the new position of P, and its coordinates x' y' z f 

 referred to the oblique axes will be OA'. OB'. OC respectively. 

 Let the unit lengths in the directions OA . OB . OC at the first 

 temperature have increased to the lengths a . ft . y in the direc- 

 tions OA x . OB'. OC x at the second; we must have 



OA^a.OA, OB'=/3.0B, 00=7-00, 

 and 



( « / + (0 + (ff = ° A2 + ° B2 + ° 2 = ° P2 ' 



If the locus of P at the first temperature be a sphere of unit 

 radius, the equation to the surface at the second temperature 

 and referred to oblique axes becomes 



But this is the well-known equation of an ellipsoid referred 

 to three conjugate diameters as axes : thus a sphere at any 

 temperature will in general become an ellipsoid at any other, 

 and any triad of perpendicular lines in the sphere will become 

 a triad of conjugate diameters of the ellipsoid; conversely, 

 any triad of conjugate diameters of the ellipsoid must have 

 been at right angles in the sphere. But one and only one 

 triad of conjugate diameters of an ellipsoid, namely the prin- 

 cipal axes, is rectangular ; and thus there must always exist 

 one and only one triad of lines which is rectangular at two 

 temperatures. If, however, the sphere remains a sphere or 

 becomes a spheroid, it is clear that it will be possible to find an 

 infinity of triads which are rectangular at two temperatures. 

 The axes of the ellipsoid will evidently be the triad of lines 

 which have experienced respectively a maximum, mean, and 

 minimum dilatation. It may also be remarked that in general 

 two, and only two, great circles of the sphere will remain 

 circles — those, in fact, which become at the second tempera- 

 ture the circular sections of the ellipsoid. The above is true 

 whatever the magnitude of the change, and whatever the cause 

 which produces it ; the only requisite is that all parallel equal 

 lines should remain parallel and equal. The state of the 

 crystal in the interval has not yet been considered. 



It will be convenient now to enter into a short digression 

 as to the permanency of a plane of symmetry at all tempera- 

 tures, and thus to consider the question of the permanency of 



