38 



MR CLERK MAXWELL ON 



This becomes zero when x = ± a where 



2 



a 2 - E b 2 

 5 



(41). 



If we wish to compare this case with that of a beam of finite length supported 

 at both ends and loaded uniformly, we must make the moment of bending zero 

 at the supports, and the length of the beam between the supports must therefore 

 be 2a Q . Substituting a for a in the value of P, we find 



P = 



h + k 



t 3 



-(s«:- 



3/; 2 + 2y 2 -2% + 



\b^{b-2y) 



(42). 



If we suppose the beam to be cut off just beyond the supports, and supported 

 by an intense pressure over a small area, we introduce conditions into the 

 problem which are not fulfilled by this solution, and the investigation of which 

 requires the use of Fourier's series. In order that our result may be true, we 

 must suppose the beam to extend to a considerable distance beyond the sup- 

 ports on either side, and the vertical forces to be applied by means of frames 

 clamped to the ends of the beam, as in Diagram Va, so that the stresses arising 

 from the discontinuity at the extremities are insensible in the part of the beam 

 between the supports. 



This expression differs from that given by Mr Airy only in the terms in the 

 longitudinal stress P depending on the function Y, which was introduced in 

 order to fulfil the condition that, when no force is applied, the beam is un- 

 strained. The effect of these terms is a maximum when y = -12788 b, and is 

 then equal to (h + &)314, or less than a third of the pressure of the beam and 

 its load on a flat horizontal surface when laid upon it so as to produce a uniform 

 vertical pressure h + k. 



