§ 1. Statement of the Means of Examination employed in these Researches.
Circular plates which produce normal vibrations are susceptible of several modes of division; sometimes they are divided into a greater or fewer number of equal sectors, always even in number, which perform their vibrations in the same time; at other times they are divided into a greater or fewer number of concentric zones; and these two series of modes of division again may be combined together, so that the acoustic figures which result are circular lines divided into equal parts by diametrical nodal lines.
If the plate which is caused to sound is perfectly homogeneous, circular, and equal in thickness, it is obvious that in the case when the figure consists of diametrical lines only, the system which they form ought to be capable of placing itself in every direction, that is to say, that any point whatever of the circumference of the plate, being taken as the place of excitation, this single condition determines the position of the nodal figure, since the point directly put in motion is always the middle of a vibrating part. In the case of circular lines, under the conditions we have just supposed, these lines would be exactly concentric with the circumference of the plate. These results are a natural consequence of the symmetry which is supposed to exist either in the form or in the structure of the plate; but if this symmetry is deranged, it will easily be conceived that an acoustical figure composed of diametrical nodal lines ought no longer to place itself in a direction depending solely on the position of the point of excitation, and that, with regard to a figure consisting of circular lines, these lines ought to be modified, and will become, for example, elliptical or of some other more complicated form. It is thus that the system of two nodal lines which intersect each other rectangularly, can upon an elliptical plate only place itself in a single position, which is on the axes of the ellipse. There is however a second position in which this mode of division can establish itself; but then it is modified in its form, and it resembles the two branches of a hyperbola, the transverse axis of which corresponds with the greater axis of the ellipse: in this latter case, the number of vibrations is less than in the first, and more so as the axes of the ellipse differ more from each other. A similar phænomenon is observed when the same mode of division is attempted to be produced on a circular plate of brass, of very equal thickness, and in which several parallel saw-cuts have been made, penetrating only to a small distance from the surface: one of the crossed nodal lines always corresponds to a saw-cut which has been made in the direction of a diameter, and the system of the two hyperbolic lines arranges itself in such a manner that the same saw-cut becomes the conjugate axis of the hyperbola. Thus, in both cases,
