82 Mr. L. Fletcher on the Dilatation of 



of errors of experiment, these thermic axes did actually coin- 

 cide with the axes of optical elasticity ; and he therefore con- 

 cluded that there really exist for all crystals what may be 

 called rectangular " morphological axes." If this be true, it 

 follows that the oblique and anorthic systems are probably 

 mere hemihedral developments of the rhombic. About the 

 time of the publication of this paper the crossed dispersion of 

 the optic axes of borax was discovered by Herschel and Nor- 

 renberg ; and thus considerable doubt may be felt as to the 

 truth of Neumann's assumption that the luminiferous aether 

 and therefore also the matter of every crystal are symmetrical 

 to three rectangular planes. Angstrom (Pogg. Ann. vol. lxxxvi. 

 1852) next made a series of experiments on gypsum and fel- 

 spar, and enunciated as result that the axes of optical elasticity, 

 of cohesion, of ordinary elasticity, and of heat expansion did 

 not all coincide, and that therefore the idea of rectangular 

 morphological axes must be given up. Grailich and Lang 

 {Sitzungsber. d. Akad. in Wien, vol. xxxiii. 1859), in their 

 celebrated memoir on the physical relations of crystals, treat 

 of this subject. They come to the conclusion that, for all 

 changes of temperature, the parametral ratios are permanently 

 either rational or irrational, and that in an oblique or anor- 

 thic crystal rectangular thermic axes do not necessarily, and 

 in fact in gypsum do not, exist — thermic axes being defined to 

 be " such directions in crystals of the non-tesseral systems as 

 at every temperature are equally inclined to each other and 

 present the same unaltered relation to the crystallographic 

 elements." It was, however, soon pointed out that an expres- 

 sion which they had found on substitution of the data for 

 gypsum to become imaginary, could in the most general case 

 be expressed as the square root ; of the sum of the squares of 

 two real quanities, and therefore was necessarily real. It is 

 clear that some numerical error had been made in their 

 calculation, and that their formulas really prove that in the 

 symmetry-plane of an oblique crystal two lines can be 

 found, at any one temperature, which are at right angles at a 

 second. It must be remarked that it is not proved that the 

 lines are at right angles at the intermediate temperatures. It 

 may be here stated that the relations given by Grailich and 

 Lang, as connecting the angles between any five planes, are 

 only proved for the case where two of the planes are crystalloid- 

 planes and the other three are zone-planes, and so can scarcely 

 be called general relations between five crystalloid-planes. If, 

 however, we remark, as will be shown later, that the constancy 

 of the indices, on which the proof depends, still holds even if 

 they be irrational, it follows that the relations mentioned above 



