140 FELIX SAVART’S RESEARCHES ON THE 
§ 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 dia- 
metrical nodal lines. 
If the plate which is caused to sound is perfectly homogeneous, cir- 
cular, 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 con- 
ditions we have just supposed, these lines would be exactly concentric 
with the circumference of the plate. These results are a natural conse- 
quence 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 how- 
ever a second position in which this mode of division can establish it- 
self; 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 vibra- 
tions is less than in the first, and more so as the axes of the ellipse differ 
more from each other. A similar phenomenon 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, 
