Page:Scientific Memoirs, Vol. 1 (1837).djvu/158

146 that, in reference to the transversal motion, the numbers of the vibrations are as the square roots of the resistance to flexion, or, which is the same thing, that the resistance to flexion is as the square of the number of oscillations.

Fig. 6 shows the results of an experiment of this kind which was made upon the same piece of beech-wood from which I cut all the plates which I shall mention hereafter. In this figure I have, to impress the mind more strongly, given to these rods directions parallel to the edges $$A X$$, $$A Y$$, $$A Z$$ of the cube fig. 5, and I have supposed that the faces of the rods are parallel to those of the cube. It is to be remarked that two sounds may be heard for the same mode of division of each rod, according as it vibrates in $$ab$$ or $$cd$$; but when they are very thin the difference which exists between them is so slight that it may be neglected. The inspection of fig. 6 shows, therefore, that the resistance to flexion is the least in the direction $$A Z$$, and is such, that being represented by unity, the resistance in the direction $$A Y$$ becomes 2.25, and 16 in the direction of $$A X$$. It is evident that the elasticity in any other direction must be always intermediate to that of the directions we have just considered.

This being well established, we shall proceed to the examination in detail of the different series of plates we have mentioned above.

Plates taken round the axis $A Y$ and perpendicular to the face $AXBZ$ of the cube.

In the plates of this series, one of the modes of division remains constantly the same. (See figs. 5, 7 and 8.) It consists of two lines crossed rectangularly, one of which, $$a y$$, places itself constantly on the axis $$A Y$$ of mean elasticity; but although this system always presents the same appearance, it is not accompanied, for the different inclinations of the plates, by the same number of vibrations; this ought to be the case, since the influence of the axis of greatest elasticity ought to be more sensible as the plates more nearly approach containing it in their plane: the sound of this system ought therefore to ascend in proportion as the plates become more nearly parallel to the plane $$C Y A X$$. As to the hyperbolic system, it undergoes remarkable transformations, which depend on this circumstance, that the line $$a y$$ remaining the axis of mean elasticity in all the plates, the line $$c d$$, which is the axis of least elasticity in No. 1, transforms itself gradually into that of the greatest elasticity, which is contained in the plane of the plate No. 6. It hence follows that there ought to be a certain degree of inclination for which the elasticities, according to the two directions $$ay$$, $$c d$$, ought to be equal: now, this actually happens with respect to the plate No. 3; and this equality may be proved by cutting in this plate, in the direction of $$a y$$ and its perpendicular, two small rods of the same dimensions: it