Page:Scientific Memoirs, Vol. 1 (1837).djvu/162

150 plane; also, the plate No. 4, fig. 10, which passes through one of the diagonals $$X Y$$ or $$A C$$, and which is perpendicular to the plane $$C Y A X$$, contains also $$A E$$ in its plane; and lastly, the plate No. 3 of fig. 12, parallel to the plane $$A D E X$$, is circumstanced in the same manner. Thus, if $$rst$$, fig. 15, is a plane perpendicular to the diagonal $$A E$$, and if the lines 1, 3, 5 indicate the directions of the three plates we have just spoken of, in order to become acquainted with the progress of the transformations which connect the modes of division of these plates together, it will be sufficient to take round $$A E$$, the projection of which is in $$c$$, a few other plates such as 2, 4, 6. The Nos. 1, 2, 3 of fig. 16 represent this series thus completed, and the dotted line $$a e$$ indicates in all the direction of the diagonal of the cube.

The nodal syytem represented by the unbroken lines consists, for No. 1, of two crossed nodal lines, one of which, $$a y$$, places itself upon the axis $$A Y$$, and the other in a perpendicular direction it transforms itself in No. 2 into hyperbolic curves, which by the approximation of their summits again become straight lines in No. 3, which contains the axis $$A Y$$ of greatest elasticity: these curves afterwards recede again, No. 4, and in the same direction as No. 2; they then change a third time into straight lines in No. 5, which contains the axis $$A Z$$ of least elasticity; and lastly, they reassume the appearance of two hyperbolic branches in No. 6.

The transformations of the dotted system are much less complicated, since it appears as two straight lines crossed rectangularly in No. 1, and afterwards only changes into two hyperbolic branches, which continue to become straighter until a certain limit, which appears to be at No. 3, and the summits of which afterwards approach each other, Nos. 5 and 6, in order to coalesce again in No. 1.

As to the general course observed by the sounds of the two nodal systems, it is very simple, and it was easy to determine it previously. Thus, the plate No. 5, containing in its plane the axis $$A Z$$ of least elasticity, the two gravest sounds of the entire series is heard; these sounds afterwards gradually rise until No. 3, which contains the axis $$A X$$ of greatest elasticity; after which they redescend by degrees in Nos. 2 and 1, (the latter contains the axis $$A Y$$ of intermediate elasticity in its plane,) and they return at last to their point of departure in the plates Nos. 6 and 5.

The transformations of the nodal lines of this series, by establishing a link between the three series of plates cut round the axes, makes us conceive the possibility of arriving at the determination of nodal surfaces, which we might suppose to exist within bodies having three rectangular axes of elasticity, and the knowledge of which might enable us to determine, a priori, the modes of division of a circular plate inclined in any manner with respect to these axes. But it is obvious, that to attempt such an investigation it would be necessary to base it on