on Change of Temperature. 293 
position of P, referred to the lines 01/, QW , the correspond- 
ing positions of the crystal-lines OL, OM, then, as before, 
0N'=(l + 8) ON=(l + o-)OPcos0, 
and -p, w = (1 + S x ) PlsT = (1 + S x ) OP sin 9. 
Whence, neglecting squares of small quantities, 
0P /2 = 0P 2 (1 + 28 cos 2 + 28, sin 2 0) ; 
andif 0P'=(1 + A)0P, 
A = Scos 2 + 6\sin 2 0, 
or A-S=(S 1 -5)sin 2 ^. 
Prop. XV. To determine the expansion perpendicular to 
the symmetry-plane of an Oblique crystal (fig. 6). 
If, at the two temperatures, bm, bm' be the angular dis- 
tances from b of a pole not lying in the plane of symmetry, it 
was shown (Prop. XIII.), from the fact of bh remaining per- 
manently a quadrant, that, omitting small quantities of the 
second order, 
bm! — bm = (8 2 — A) sin bm sin mh, 
A being the coefficient of expansion of Oh, the line of inter- 
section of the zone-plane [bm'] with the plane of symmetry, 
and S 2 the coefficient of expansion of Ob, a line perpendicular 
to the plane of symmetry. 
If be the inclination of OA to that thermic axis of which 
the coefficient of expansion is 8, then (Prop. XIV.) 
A -8 = (8,-8) sin 2 0, 
whence 
sin bm sin mh 
j, rj bm' —bm , ,„ ^ . 9 
6 2 — 8 — -s— = : + (S\ — 8) sm 2 1 
** Om /-nil cm m #i x ' 
We shall now illustrate the above formulae by a practical 
application of them. 
The only published measurements of the angles of an 
Oblique crystal at different temperatures were, until the recent 
determinations of Beckenkamp, those made upon gypsum by 
Mitscherlich, the discoverer of the angular variations produced 
in crystals by change of temperature. From these measure- 
ments Neumann has calculated the positions and expansions of 
the thermic axes, and we shall thus be able to directly com- 
pare the results obtained by his method with those deduced 
from the preceding formulas. 
Fig 7 a represents a crystal of gypsum : b is the cleavage- 
plane and plane of symmetry (0 1 0), m m x are planes of the 
prism {11 0}, and / 1 ± of the form {111}; the planes a (1 0), 
