Page:Scientific Memoirs, Vol. 1 (1837).djvu/272

260 the same mode of division in No. 4, perpendicular to the face $$a X b$$ of the pyramid. Lastly, from No. 11 until the plate perpendicular to the axis, the sounds approximate again, as well as the summits of the hyperbolic curves, and at the same time the two systems of nodal lines again become rectangular; the sounds thus become almost the same.

Among the plates which we have just examined there are two which merit particular attention; these are Nos. 5 and 11, parallel to the the faces $$eXd$$ and $$aXb$$ of the pyramid, and the elastic state of which undoubtedly differs very much, since in one it is the hyperbolic system which gives the gravest sound, whilst in the other it is the rectangular system, and that, besides, there is a great difference between the sounds which correspond to each of their nodal systems. The faces $$aXb$$ and and $$eXd$$ of the pyramid being opposite, one of the two ought to be susceptible of cleavage, whilst the other ought not to be capable of this mechanical division; consequently if we knew which of the two plates Nos. 5 and 11 possesses this property, we might, by examining its acoustic figures, determine which are the faces of the pyramid parallel to the faces of the primitive rhombohedron. Rock crystal not yielding in the least to any attempt at dividing it into regular layers in any direction, it was impossible for me to ascertain directly which of the two faces $$aXb$$ or $$eXd$$ were those of cleavage; but this question can be resolved with ferriferous carbonate of lime, a substance which is cleaved with almost the same facility as pure carbonate of lime, and which appears to possess, in reference to sonorous vibrations, properties in general analogous to those of rock crystal. Now, if we cut in such a crystal two plates,—one taken parallel to a natural face of the rhombohedron, the other corresponding with a plane inclined to the axis by the same number of degrees as these faces, and which are besides equally inclined to the two faces which form one of the obtuse solid angles,—we find that the first possesses the same properties as No. 11, whilst the second has a structure analogous to that of No. 5; whence it ought to be concluded, from analogy, that the face $$aXb$$ of the pyramid fig. 1. is that which is susceptible of cleavage. This once established, it is not even requisite, in order to ascertain which of the faces is susceptible of cleavage, to cut a plate parallel to one of these faces; it is obvious that a plate parallel to the axis and normal to two parallel faces of the hexahedron should be sufficient to attain this end. Thus, let fig. 5, $$a b c d e f$$, be the horizontal projection of the prism represented fig. 1; according to what has been said, $$r s t v$$ will be the projection of the primitive rhombohedron; again, let $$ll^\prime$$ be the projection of a plate parallel to the axis and equally inclined to the two faces of $$a$$ and $$f$$ of the hexahedron; according to what we have above said, this plate will assume the mode of division of No. 3, fig.2, bis, and the line $$op$$ will be parallel to the plane $$r stu$$ normal to the plate, that is to say,