146 : FELIX SAVART’S RESEARCHES ON THE 
that, in reference to the transversal motion, the numbers of the vibra- 
tions are as the square roots of the resistance to flexion, or, which is 
the same thing, that the resistance to flexion is as the square of the 
number of oscillations. 
Fig. 6 shows the results of an experiment of this kind which was 
made upon the same piece of beech-wood from which I cut all the 
plates which I shall mention hereafter. In this figure I have, to impress 
the mind more strongly, given to these rods directions parallel to the 
edges A X, A Y, AZ of the cube fig. 5, and I have supposed that the 
faces of the rods are parallel to those of the cube. It is to be remarked 
that two sounds may be heard for the same mode of division of each 
rod, according as it vibrates in a 6 or ed; but when they are very thin 
the difference which exists between them is so slight that it may be 
neglected. The inspection of fig. 6 shows, therefore, that the resistance 
to flexion is the least. in the direction A Z, and is such, that being re- 
presented by unity, the resistance in the direction A Y becomes 2°25, 
and 16 in the direction of A X. It is evident that the elasticity in any 
other direction must be always intermediate to that of the directions we 
have just considered. 
This being well established, we shall proceed to the examination in 
detail of the different series of plates we have mentioned above. 
First Sertes.—Plates taken round the axis AY and perpendicular to 
the face AX BZ of the cube. 
In the plates of this series, one of the modes of division remains con- 
stantly the same. (See figs. 5, 7 and 8.) It consists of two lines crossed 
rectangularly, one of which, a y, places itself constantly on the axis A Y 
of mean elasticity; but although this system always presents the same 
appearance, it is not accompanied, for the different inclinations of the 
plates, by the same number of vibrations; this ought to be the case, 
since the influence of the axis of greatest elasticity ought to be more 
sensible as the plates more nearly approach containing it in their plane: 
the sound of this system ought therefore to ascend in proportion as the 
plates become more nearly parallel to the plane CY AX. As to the © 
hyperbolic system, it undergoes remarkable transformations, which de- 
pend on this circumstance, that the line a y remaining the axis of mean 
elasticity in all the plates, the line ed, which is the axis of least elasti- 
city in No. 1, transforms itself gradually into that of the greatest elas- 
ticity, which is contained in the plane of the plate No.6. It hence 
follows that there ought to be a certain degree of inclination for which 
the elasticities, according to the two directions ay, ¢ d, ought to be 
equal: now, this actually happens with respect to the plate No. 3; and 
this equality may be proved by cutting in this plate, in the direction 
of ay and its perpendicular, two small rods of the same dimensions: it 
