Mr. Rankine on Axes of Elasticity and Crystalline Forms. 303 



variables called Contra-ordinates. The contra-ordinates of a point 

 R are the projections of the radius-vector OR on the three axes. 

 For rectangular axes, co-ordinates and contra-ordinates are identical. 

 The co-ordinates x, y, z and contra-ordinates u, v, w of a point R 



are connected by the equation 



ux-j-vy + wz^OK^. 

 As there are six independent quantities in the directions of a 

 system of three axes of indefinite obliquity, there is a system of 

 right or oblique axes in every solid for which six of the coefficients 

 of the characteristic function of the Biquadratic surface disappear, 

 reducing that function to its canonical form of nine terms. These 

 three axes are called the 



Principal Euthttatic Axes. 



Every Euthytatic axis has this property, that a direct linear elonga- 

 tion or compression along such an axis, jjroduces a normal stress, and 

 no oblique or tangential stress on a plane normal to the same axis. 



Every Euthytatic axis is normal to the Biquadratic Surface, and 

 is a line along which the direct elasticity of the body is either a 

 maximum or a minimum, or in that condition which combines the 

 properties of maximum and minimum. 



It is probable that the faces or edges of jmmitive crystalline forms 

 are normal to Eiithytatic axes, and that the planes of cleavage in 

 crj'stals are normal to Euthytatic axes of minimum elasticity. 



It is also probable that the symmetrical summits of crystals cor- 

 respond to Euthytatic axes. 



There are, in every solid, at least the three principal Euthytatic 

 Axes, which are normal to the faces of a hexahedron, right or oblique 

 as the case may be. In certain cases of symmetry of these axes 

 and of the Homotatic Coefficients, there are Secondary or Additional 

 Euthytatic Axes, which are determined by the following principles : 



1 . When the three principal axes and the Homotatic Coefficients 

 are symmetrical round a central axis, that axis is an additional 

 Euthytatic axis. , 



2. When there are a pair of orthogonal Euthytatic axes in a given 

 plane, there may be, under certain conditions sj)ecified in the paper, 

 a pair of additional or secondary axes in that plane, making with 

 each other a pair of angles bisected by the orthogonal axes. 



In the first column of a table, the possible systems of Euthytatic 

 axes are arranged according to a classification and nomenclature of 

 their degrees and kinds of symmetry ; and in the second and third 

 columns are stated the primitive crystalline forms to the faces and 

 edges of which such systems of axes are respectively normal, and 

 which embrace all the primitive forms known in crystallography. 



The six Heterotatic Coefficients are independent of the fifteen 

 Homotatic Coefficients which determine the Efithytatic axes. 



The paper concludes witii observations on some real and alleged 

 differences between the laws of solid elasticity and those of the 

 luminiferous force, — on some hypotheses in connexion with the 

 wave-theory of light, — and on the refraction of light in crystals as 

 connected with the symmetry of their Euthytatic axes. 



