Page:Scientific Memoirs, Vol. 1 (1837).djvu/160

148 It is almost useless to observe that the plates taken in the directions I, II, III, inclined on the other side of the axis $$A X$$ the same number of degrees as the plates 1, 2, 3, would present exactly the same phænomena as these latter. This observation being equally applicable to the following series, we shall not mention it again.

— Plates taken round the axis $A Z$ of least elasticity and perpendicular to the plane $C Y A X$; figs. 9 and 10.

As in the preceding case, one of the nodal systems of the plates of this series consists of two lines crossed rectangularly, one of which, $$a z$$, corresponds with the axis $$A Z$$; whence it follows that the second may be considered as the projection of the two other axes on the plane of the plate, which, whatever its inclination may be, ought consequently to possess a greater elasticity in the direction $$f g$$ than in the direction $$a z$$: thus the hyperbolic system of this series cannot present the transformations which we saw in the preceding series, where $$cd$$, fig. 8, possesses sometimes a less, at other times a greater elasticity than that of $$a y$$. In the present case, $$a z$$ remaining constantly the axis of least elasticity, the resistance to flexion in the direction $$f g$$ goes on gradually increasing from the plate No. 1 to the plate No. 6 parallel to the plane $$A X B Z$$, and the branches of the hyperbola straighten themselves in proportion as the plates more nearly approach this last position. As to the sounds which correspond to each of these nodal systems, it is observed that they ascend gradually from No. 1 to No. 6, and that the sound of the hyperbolic system is sharper in a part of the series than that of the system of crossed lines, whilst they become graver in the other part. There is therefore a certain inclination for which the sounds of the two systems ought to be equal; and this evidently would have taken place in the present experiment for a plate intermediate to No. 4 and No. 5.

The interval between the gravest and the sharpest sound of each series was an augmented fifth.

— Plates taken round the axis $A X$ of greatest elasticity, and perpendicular to the plane $A Y D Z$; figs. 11 and 12. The elastic state of these plates cannot present such remarkable differences as those we have observed in the preceding series; for, being all cut round the axis of greatest elasticity, they can only contain in their plane that of least or that of mean elasticity, or lastly, those intermediate between these limits, which do not vary greatly from each other. Thus it is seen that their modes of division differ very little from each other, and that the sounds which correspond to them present rather slight differences, although they go on ascending in proportion as the plates more nearly approach containing the axis of mean elasticity in their plane. Here, as in the other series, one of the nodal