276 Mr. L. Fletcher on the Dilatation of Crystals 
geometrical and physical characters of a crystal are the same 
along all lines having the same direction : — 
1. A plane of physical and geometrical symmetry is perma- 
nent for all changes of temperature so long as the crystalline 
arrangement is preserved. A crystal may thus, theoretically 
at least, pass on mere change of temperature from a system 
of lower to one of higher symmetry', but not vice versa, with- 
out a preliminary destruction of the crystalline arrangement 
of its molecules. 
2. A sphere at one temperature in general becomes an 
ellipsoid at a second. 
3. Any three lines mutually perpendicular at the first tem- 
perature become three conjugate diameters of the ellipsoid at 
the second. 
4. One particular triad of lines, namely that which at the 
second temperature coincides with the axes of the ellipsoid, is 
rectangular at both temperatures : the lines of this triad are 
the directions of greatest mean and least expansion, and have 
been called by Neumann thermic axes. 
5. The pair of diameters at right angles to the circular 
sections of the ellipsoid are such that the expansion of all 
crystal-lines normal to them at the second temperature has 
been the same : they thus present a character analogous to 
that of the optic axes and of the magnetic axes of Plucker. 
6. One triad of lines has absolutely the same position in 
space at the two temperatures, and at the suggestion of Prof. 
Maskelyne has been designated as a triad of atropic lines ; 
being in general not at right angles, they are distinct from 
the thermic axes ; and if permanently fixed in space for all 
variations of temperature they must be intimately related to 
the structure of the crystal, and would in such case have a 
claim to be regarded not as arbitrary but as veritable axes of 
the crystal. 
7. An infinite number of triads of lines can be found 
which retain their mutual inclinations while changing their 
directions in space: the lines of such triads will in this paper be 
designated as isotropic ; the so-called thermic axes are merely 
the particular triad of isotropic lines for which the angles of 
mutual inclination are right angles. 
8. If in a crystal belonging to the Oblique system the two 
lines in the symmetry-plane which are atropic for one pair of 
temperatures are atropic for all others, no lines isotropic for 
one pair of temperatures will retain their inclination perma- 
nently, and thus having no constancy of inclination can scarcely 
be important as structural lines of the crystal. 
It was further shown that in the Cubic system all lines, and 
