Page:Scientific Memoirs, Vol. 1 (1837).djvu/153

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the transverse axis of the hyperbola is always in the direction of the least resistance to flexure.

Let us now suppose that, the plate remaining perfectly circular and of equal thickness, it possesses in its plane a degree of elasticity which is not the same in two directions perpendicular to each other; the symmetrical disposition round the centre being then found to be destroyed, although in another manner than in the two examples we have just adduced, an analogous result ought still to be obtained.

Thus, if we take a plate of this description, a plate of wood, for instance, cut parallel to the fibres, and fixing it lightly by its centre, endeavour to make it produce the mode of division consisting of two lines crossed rectangularly, we shall find that when it thus divides itself, the lines of rest always place themselves according to the directions of the greatest and least resistance to flexure, and that putting it afterwards in motion at the extremity of the preceding lines, it may be made to produce a second mode of division, which presents itself under the aspect of a hyperbola the branches of which are much straightened, and which would have for its conjugate axis that line of the cross which corresponds to the direction of the greatest resistance to flexure. In short, when the symmetrical disposition round the centre is destroyed, no matter in what way, the mode of division formed by two nodal lines which intersect each other rectangularly can place itself only in two determinate positions, for one of which it presents frequently the appearance of two hyperbolic branches more or less straightened; and, as we shall soon see, it may even happen that, for certain distributions of elasticity, this mode of division presents itself under the form of two hyperbolic curves in the two positions in which it becomes possible. Lastly, if a similar plate be caused to produce some of the high modes of division, but yet consisting of diametrical lines, experiment shows that they can likewise place themselves in two invariable positions, and pass through certain modifications analogous to those which the system of two lines crossed at right angles undergoes. Thus the immoveability of the nodal figures, and the double position which they can assume, are distinctive characters of circular plates all the diameters of which do not possess a uniform elasticity or cohesion.

It follows therefore from the preceding, that by forming with different substances circular plates of very equal thickness, we may, by the fixed or indeterminate position of an acoustic figure consisting of diametrical nodal lines, ascertain whether the properties of the substance in question are the same in all directions. By applying this mode of examination to a great number of plates formed of different substances regularly or confusedly crystallized, as the metals, glass, sulphur, rock-crystal, carbonate of lime, sulphate of lime, gypsum, &c., it is constantly found that the acoustic figure, formed of two lines crossed rectangularly, can only place