Page:Scientific Memoirs, Vol. 1 (1837).djvu/159

Rh will be seen, on causing them to vibrate in the same mode of transversal motion, that they produce the same sound. It also follows, because the elasticity in the direction $$a y$$ is sometimes smaller and sometimes Greater than that which exists in the direction of $$c d$$, that the first axis of the nodal hyperbola ought to change its position to be able to remain always perpendicular to that of the lines $$a y$$, $$c d$$, which possess the greatest elasticity; thus, in Nos. 1 and 2, $$c d$$ possessing the least elasticity, it becomes the transverse axis of the hyperbola, whilst in Nos. 4, 5 and 6, the elasticity being greater in the direction c d than in that of $$a y$$, the transverse axis of the hyperbola places itself on the latter line. As the ratio of the two elasticities varies only gradually, it is obvious that the modifications impressed on the hyperbolic system ought in the same manner to be gradual: thus the summits of these curves, at first separated in No. 1 by a certain distance (which will depend on the nature of the wood), will approach nearer and nearer, for the following plates, until they coalesce as in No. 3, at a certain degree of inclination, which was 45° in the experiment to which I now refer, but which might be a different number of degrees for another kind of wood. At the point where we have seen that the elasticities are equal in the direction of the axis, the two curves transform themselves into two straight lines which intersect each other rectangularly, after which they again separate; but their separation is effected in a direction perpendicular to that of their coalescence. The sounds of the hyperbolic system follow nearly the same course as those of the system of crossed lines, that is to say, they become higher in proportion as the plates more nearly approach being parallel to the axis of greatest elasticity; but it deserves to be remarked, that the plate No. 3, for which the elasticity is the same in the two directions $$a y$$, $$c d$$, is that between the two sounds [of which there is the greatest interval: this evidently depends on the elasticity in the two directions $$a y$$, $$c d$$ being very different from that which exists in the other directions of the plate.

Lastly, it is to be remarked that, in the four first plates, the sound of the hyperbolic system is sharper than that of the system of crossed lines, and that it is the contrary for the plate No. 6, which renders it necessary that there should be between No. 4 and No. 6 a plate, the sounds of which are equal, which in the present case is exemplified in No. 5, although its two modes of division differ greatly from each other. There is another thing remarkable in this plate; its two modes of division can transform themselves gradually into each other by changing the position of the place of excitation, so that the two points $$c$$ and $$c'$$ becoming two nodal centres, are in every respect in the conditions indicated by fig. 4.

The interval included between the gravest and the sharpest sounds of this series was an augmented sixth.