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424                             ON VIBRATIONS  AND  DEFLEXIONS  OF                            [411
that B&wldx* is a linear function of as between 0 and 6, and again a linear function between b and I, the two linear functions assuming the same value at x = b. Since B is everywhere positive, it follows that the curvature cannot vanish more than twice in the whole range from 0 to I, ends included, unless indeed it vanish everywhere over one of the parts. If one end be supported, the curvature vanishes there. If the other end also be supported, the curvature is of one sign throughout, and the curve of deflexion can nowhere cross the axis. If the second end be clamped, there is but one internal point of inflexion, and again the axis cannot be crossed. If both ends are clamped, the two points of inflexion are internal, but the axis cannot be crossed, since a crossing would involve three points of inflexion. If one end be free, the curvature vanishes there, and not only the curvature but also the rate of change of curvature. The part of the rod from this end up to the point of application of the force remains unbent and one of the linear functions spoken of is zero throughout. Thus the curvature never changes sign, and the axis cannot be crossed. In this case equilibrium requires that the other end be clamped. We conclude that in no case can there be a deflexion anywhere of opposite sign to that of the force applied at x  6, and the conclusion may be extended to a force, however distributed, provided that it be one-signed throughout.
Leaving the problems presented by the membrane and the bar, we may pass on to consider whether similar propositions are applicable in the case of a flat plate, whose stiffness and density may be variable from point to point. An argument similar to that employed for the membrane shows that when the boundary is clamped any contraction of it is attended by a rise of pitch. But (Theory of Sound,  230) the statement does not hold good when the boundary is free.
When a localized transverse force acts upon the plate, we may inquire whether the displacement is at all points in the same direction as the force. This question was considered in a former paper* in connexion with a hydrodynamical analogue, and it may be convenient to repeat the argument. Suppose that the plate (fig. 2), clamped at a distant boundary, is almost divided into two independent parts by a straight partition CD extending across, but perforated by a narrow aperture AB\ and that the force is applied at a distance from CD on the left. If the partition were complete, w and dw/dn would be zero over the whole (in virtue of .the clamping), and the displacement in the neighbourhood on the left would be simple one-dimensional bending, with w positive throughout. On the right w would vanish. In order to maintain this condition of things a certain couple acts upon the plate in virtue of the supposed constraints along CD.
* Phil. Mag. Vol. xxxvr. p. 354 (1893); Scientifa Papers, Vol. iv. p. 88.
?. 2.that at which it usually stands, on account oi the pressure of the atmosphere. If the mercury has been well boiled in the tube, it may be made to remain in contact with the closed end, at the height of 70 inches or more " (Young's Lectures, p. 626,1807). If the mercury be wet, boiling may be dispensed with and negative pressures of two atmospheres are easily demonstrated.