directions perpendicular to each other; this explains the production of the nodal lines crossed at right angles, as in the series of plates cut round one of the axes of elasticity, in bodies in which these axes are rectangular. It appears therefore that we may conclude from this observation that rock crystal possesses, like carbonate of lime, supernumerary planes of cleavage parallel to the diagonal planes of its primitive rhombohedron, and that it is to the existence of these supernumerary joints that the principal peculiarities of the elastic state of this substance must be attributed.
The only striking difference there appears to be between the structure of carbonate of lime and that of quartz consists in this, that, in the first of these substances, the small diagonal of the rhombohedron is the axis of least elasticity, whilst it is that of greatest elasticity in the second. To be convinced of the accuracy of this assertion, it is sufficient to cut, in a rhombohedron of carbonate of lime, a plate taken parallel to one of its natural faces, and to examine the arrangement of its two nodal systems, one of which consists of two lines crossed rectangularly, which are always placed on the diagonals of the lozenge, the primitive outline of the plate, and the other is formed of two hyperbolic branches, to which the preceding lines serve as axes (see fig. 7, bis, No. 6); but with this peculiarity, that it is the small diagonal which becomes the first axis of the hyperbola, whilst it is its second axis in the corresponding plate of rock crystal (see fig. 3, bis, No. 11). It may be here asked how far this difference of structure may influence the phænomena of light which are peculiar to each of these two substances, one of which is a crystal with attractive (positive) double refraction, and the other with repulsive (negative) double refraction.
It appears, therefore, to result from this approximation between the phænomena presented by carbonate of lime and rock crystal, with respect to sonorous vibrations, that the arrangement of the acoustic figures, and the numbers of vibrations by which they are accompanied, are always found intimately connected with the directions of cleavage in each plate; and it may be said in general, that if these directions intersect each other at right angles, in the plane of the plate, one of the two modes of division will always consist of two lines crossed rectangularly; whilst if they are inclined to each other the two nodal systems will be hyperbolic curves.
The disposition of the nodal lines upon circular plates of sulphate of lime gives additional support to tliis conclusion. For thin plates of this substance break according to two directions inclined to each other at 113° 8′; and experiment shows that the two modes of division of which they are susceptible are two nearly similar hyperbolic curves, one of which appears to have for its asymptotes the directions of cleavage, and the other for its principal axis that one of these two directions in
