150 FELIX SAVARt's RESEARCHES ON THE 



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 m ith 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 S5'stem represented by the unbroken lines consists, for 

 No. 1, of two crossed nodal lines, one of which, ai/, places itself upon 

 the axis A Y, and the other in a perjjendicular direction ; it transforms 

 itself in No. 2 into hj^erboUc curves, which by the approximation of 

 their summits again become straight lines in No. 3, which contains the 

 ^ixis 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 transfonnations 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 con- 

 tinue to become straighter until a certain limit, which appears to be at 

 No. 3, and the summits of which afterwards approach each other, 

 Nps. 5 and 6, in order to coalesce again in No. 1 . 



As to the general course observed by the sounds of the two nodal sy- 

 stems, it is very simple, and it was easy to determine it previously. Thus, 

 the plate No. 5, containing in its pleuie 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 tliree series of plates cut I'ound the axes, makes us 

 conceive the possibility of arriving at the determination of nodal sur- 

 faces, which we might suppose to exist within bodies having'three rect- 

 angular axes of elasticity, and the knowledge of wliich might enable 

 us to determine, d priori, the modes of division of a circular plate in- 

 clined 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 



