AND DIAGRAMS OF FORCES. 205 



If the total longitudinal stress across any vertical section of the beam is zero, 



(IF 

 the value of -j- must be the same when y = and when y = b. From this 



condition we find the value of P by equation (32) 



(39). 



f 



J 



The moment of bending at any vertical section of the beam is 



(40). 



This becomes zero when x= a a where 



o S = 2 -F (41). 



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

 ported 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 . Substituting a for a in the value of P, we find 



(42). 



If we suppose the beam to be cut oft just beyond the supports, and sup- 

 ported 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 supports on either side, and the vertical forces to be applied by means 

 of frames clamped to the ends of the beam, as in Diagram V, 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 

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

 then equal to (h + k) '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 



