on Change of Temperature. 277 
in the Tetragonal and Hexagonal systems all lines parallel or 
perpendicular to the axis of highest symmetry, are permanent 
in direction on change of temperature of the crystal ; that in 
the Rhombic system both the thermic axes and the atropic 
lines are coincident with the crystallographic axes ; that in 
the Oblique system one thermic axis and one atropic line 
coincide with the crystallographic axis perpendicular to the 
plane of symmetry, while the remaining thermic axes, atropic 
lines, and crystallographic axes lie without any evident con- 
nection in the plane of symmetry ; and that in the Anorthic 
system the thermic axes, atropic lines, and crystallographic 
axes are apparently quite independent of each other. 
With the view of experimentally testing the above theory 
of the possible want of permanency of the thermic axes, Mr. 
Beckenkamp*, at the instance of Professor Groth, has under- 
taken in the Strassburg laboratory the onerous task of making 
precise measurements of the angles of anorthite, axinite, ortho- 
clase, and gypsum at various temperatures; and for this pur- 
pose a large goniometer reading to seconds has been at his 
disposal. Deducing from these measurements the positions 
of the thermic axes by the help of formulas constructed by 
the elder and younger Neumann for Oblique and Anorthic 
crystals respectively f, Mr. Beckenkamp finds that, while the 
variation in the case of the other minerals examined is within 
the limits of the errors of experiment, in a crystal of anorthite 
the axis of minimum expansion for the pair of temperatures 
20° — 80° C. is as much as 2Q° 3 & distant from the corresponding 
axis for the pair of temperatures 20° — 200° C. ; and Mr. Becken- 
kamp further states that it is unnecessary to show that this is 
beyond any possible experimental error. 
Any deviation, however small, from absolute rectangularity 
of the lines which are thermic axes for one pair of tempera- 
tures being clearly fatal to their permanency, the magnitude 
of this deviation was not touched upon in the first paper. As, 
however, the change in the mutual inclinations of these lines 
could scarcely be expected to exceed a few seconds, and the 
change in the position of the thermic axes perhaps a few 
minutes, it seemed necessary to carefully examine the original 
formulae of Neumann and the observations and calculations 
of Beckenkamp before such an enormous variation as 26° 36' in 
the position of a thermic axis could be considered absolutely 
* Zeitschrift fur Krystallographk, 1881, vol. v. p. 436 ; and 1882, vol. vi. 
p. 450. 
t Pogg. Ann. 1833, vol. xxvii. p. 240 ; and 1861, vol. cxiv. p. 492. 
