344 Mr. L. Fletcher on the Dilatation of Crystals 
the liquid near the surface, which he ignores, that the surface- 
tension exists, and that the surface layers are a seat of energy, 
and physically different from the liquid in the interior. 
I am, Gentlemen, 
Your obedient Servant, 
Clifton, October 1, 1883. A. M. WoKTHINGTON. 
XL VIII. The Dilatation of Crystals on Change of Temperature. 
By L. Fletchek, M.A., of the Mineral Department, British 
Museum; late Felloio of University College, Oxford. 
[Continued from p. 300.] 
Second Part. 
ALTHOUGH we can thus find to any degree of accuracy 
the isotropic planes and the maximum and minimum 
expansions for lines in any zone-plane of an Anorthic crystal, 
and therefore the positions and relative magnitudes of the 
principal axes of the section of the ellipsoid made by the given 
zone-plane, the above method cannot be satisfactorily extended 
to the calculation of the directions and relative magnitudes of 
the three principal expansions of the crystal. 
We shall accordingly solve the problem in an entirely 
different way, and, to render the solution for three dimensions 
more easy to follow, shall take first the case of an Oblique 
crystal and obtain remarkably simple formulae for the calcu- 
lation of the thermic axes directly from the parameters them- 
selves. 
At the first temperature let OA, OB, OC be the crystallo- 
graphic axes, and let their lengths be denoted by A, B, C ; 
as usual let OB be the axis perpendicular to the plane of 
symmetry containing the two axes OA, OC. At the second 
temperature the axis OB' will coincide in direction with OB, 
but will have a length B', and the axes OA, OC will have 
taken up new directions OA', OC and new lengths A', C. 
The measurement of the angles of a crystal at two tempe- 
ratures can give only relative, not absolute, displacements of the 
lines; and without affecting calculations of these relative posi- 
tions, the crystal may be imagined to be rotated at the second 
temperature round OB until OC / coincides in direction with 
OC. If it should be hereafter necessary to determine the 
absolute displacement of any line, this imaginary rotation of 
the line with the whole crystal will have to be compounded 
with the relative displacement now about to be calculated; 
till then this rotation of the whole crystal may be disregarded. 
It will be convenient first to calculate from the alterations 
