Crystals on Change of Temperature. 83 



are true in any case, it being a matter of no moment whether the 

 planes be crystalloid-planes, zone-planes, or neither. C. Neu- 

 mann (Pogg. Ann. vol. cxiv. 1861) shows that, in an anorthic 

 crystal, for any two temperatures three lines can be found 

 which are at right angles at both, and shows how their posi- 

 tions may be calculated from the observed angles of the crystal 

 at the two temperatures. In 1868 (Pogg. Ann. vol. cxxxv.) 

 appeared a paper by C. Pape, on the " thermic and chemical 

 axes" of gypsum and chalcanthite, in which he (somewhat in- 

 accurately) states that, in the preceding article, Neumann has 

 proved the existence of axes which remain at right angles at 

 all temperatures. From his experiments he infers the coinci- 

 dence of the thermic and chemical axes in both gypsum and 

 chalcanthite, whence he rapidly passes to the belief that he has 

 established the existence of rectangular morphological axes for 

 all crystals. 



It is difficult to see how the dispersion of the planes of the 

 optic axes, met within many crystals, can be simply explained by 

 the theory of the existence of these rectangular morphological 

 axes. In the same volume of the Annalen is published one of 

 Fizeau's papers on the dilatation of crystals as determined by 

 experiment; and he finds, by his admirable method, that the 

 coefficients of dilatation of felspar along two directions equally 

 inclined to an optic mean line, and on opposite sides of it, in 

 the plane of symmetry, are almost in the ratio of 5 : 1 ; and 

 thus the belief in the existence of any simple relation between 

 the positions of the axes of dilatation and the axes of optical 

 elasticity ought to be finally discarded. In his treatment of 

 the theoretical part of the question, Fizeau assumes that the 

 directions of maximum, mean, and minimum dilatation are 

 permanent for all temperatures ; and in this assumption he is 

 followed by Groth (Physikalische Krystallographie, 1876, 

 p. 140). Summing up, we may say that up to the present 

 it has merely been proved, and that in a laborious manner, 

 that at any two temperatures there are three crystal-lines at 

 right angles at both : it has not been proved that these lines 

 are at right angles at other temperatures or are fixed in space; 

 but still Pape, Fizeau, and Groth seem to think they may 

 legitimately assume it. 



We shall assume that the physical and geometrical proper- 

 ties along all parallel lines of a crystal are the same. Hence it 

 immediately follows that any set of particles of the crystal in 

 a right line at any one temperature will be so at any other, 

 although the position in space and the length of the line may 

 vary ; equal parallel lines will remain equal and parallel ; 

 parallel planes will remain parallel planes ; parallelograms 



G2 



