THE EQUILIBRIUM OF ELASTIC SOLIDS. 57 



Let b be the breadth of the beam, then the force with which the beam 

 resists flexure = M. 



which is the ordinary expression for the stiffness of a rectangular beam. 



The stiffness of a beam of any section, the form of which is expressed by 

 an equation between x and y, the axis of x being perpendicular to the plane of 

 flexure, or the osculating plane of the axis of the beam at any point, is ex- 

 pressed by 



Mc = E ly'dx ................................. (50), 



M being the moment of the force which bends the beam, and c the radius of 

 the circle into which it is bent. 



CASE VI. , 



At the meeting of the British Association in 1839, Mr James Nasmyth 

 described his method of making concave specula of silvered glass by bending. 



A circular piece of silvered plate-glass was cemented to the opening of an 

 iron vessel, from which the air was afterwards exhausted. The mirror then 

 became concave, and the focal distance depended on the pressure of the air. 



Buffon proposed to make burning-mirrors in this way, and to produce the 

 partial vacuum by the combustion of the air in the vessel, which was to be 

 effected by igniting sulphur in the interior of the vessel by means of a burn- 

 ing-glass. Although sulphur evidently would not answer for this purpose, phos- 

 phorus might; but the simplest way of removing the air is by means of the 

 air-pump. The mirrors which were actually made by Buffon, were bent by 

 means of a screw acting on the centre of the glass. 



To find an expression for the curvature produced in a flat, circular, elastic 

 plate, by the difference of the hydrostatic pressures which act on each side 

 of it, 



Let t be the thickness of the plate, which must be small compared with 

 its diameter. 



Let the form of the middle surface of the plate, after the curvature is 

 produced, be expressed by an equation between r, the distance of any point 

 from the axis, or normal to the centre of the plate, and x the distance of 

 the point from the plane in which the middle of the plate originally was, and let 



VOL I. 



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