Page:Scientific Memoirs, Vol. 1 (1837).djvu/156

144 being capable of transforming itself into the rectangular system, when the position of the point put directly in motion is made to vary.

Examining with care the nodal lines in fig. 2, it is found equally that its two nodal systems can thus change themselves one into the other; and the same phænomenon is reproduced in the plate No. 4, in which the values of the extreme elasticities differ still more, and in which the points $$a$$ and $$b$$ recede from each other at the same time as the curves become more straightened. In the plate No. 5, parallel to the axis $$A Y$$, these curves are no longer susceptible of assuming any other position than that indicated in the figure. Thus, in No. 1, the centres $$a$$ and $$b$$ coalesce into one, and there is only a single figure consisting of two crossed lines, the system of which can assume any position; these centres afterwards gradually receding, the modes of division can change themselves from one into the other, and at last, when the branches of the curve are nearly straight lines, the two figures become perfectly fixed.

The existence of these nodal points or centres is, without doubt, a very remarkable phænomenon, and which it will be important to study with great care. In order to give an accurate idea of it, I have in fig. 4 indicated by a dotted line the successive modifications which the two hyperbolic lines assume when the plate is fixed at one of the points $$a$$ or $$b$$, and the place of excitation moves gradually from $$e$$ to $$e' e''$$, passing over a quarter of the circumference of the plate. When the motion is excited in the vicinity of $$e$$, the curves are by the union of their summits transformed into two straight lines which intersect each other rectangularly; and it is obvious that if it had been excited near $$e'$$, the two branches of the curve would re-appear, but with this peculiarity, that their transverse axis would take the position assumed by the conjugate, when the motion was produced on the other side of $$e''$$.

As to the numbers of the vibrations which correspond to each mode of division, for the different degrees of inclination of the plates, it will be seen by examining fig. 3, that, at first equal in No. 1, they go on continually increasing and receding from each other up to No. 5, which contains the axis of the cylinder; and it is indeed evident, that the elasticity in the direction perpendicular to the axis remaining the same for all the plates, whilst that which is perpendicular to this direction goes on continually increasing, this ought to be, in general, the progress of the phænomenon.

These experiments were made with plates of oak 8·4 cent. (3·3071 inches) in diameter, and 3ᵐ·7 (·1456 inch) in thickness: they were repeated with plates of beech-wood, and analogous results were obtained; only the ratio between the two elasticities not being the same, the interval between the two sounds of each plate was found to be greater.

The most general consequence that can be deduced from the