108 MR JAMES CLERK MAXWELL ON THE 
C, may be found by assuming that when r=a, p=h,, and g may be found from p 
by Equation (21.) 
As the expressions thus found are long and cumbrous, it is better to use the 
following approximations :— 



__ (9mr) y 
= (Gren) DNs Ni aT 
9mp\ 1 (e-@ 
p= Sei SS +e+y) . (48) 
In these expressions @ is half the depth of the beam, and y is the distance of 
any part of the beam from the neutral surface, which in this case is a cylindric 
surface, whose radius is ¢. 
These expressions suppose ¢ to be large compared with a, since most substances 
a : . 
break when — exceeds a certain small quantity. 
Let 6 be the breadth of the beam. then the force with which the beam resists 
flexure =M. 

hel EO GO CBE 
m= foy9 = 20h 2 3 Shee an hes ices 
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 z 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 expressed by 
Me=E fy?dz, oor, aa) 
M being the moment of the force which bends the beam, and ¢ the radius of the 
circle into which it is bent. 
Case VI. 
At the meeting of the British Association in 1839, Mr James Nasmyvu de- 
scribed 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. 
Burron 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 burning- 
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 Burron, were bent by means 
of a screw acting on the centre of the glass. 
