304 Dr. J. Prescott on the 



Case 4. — The beam carries a load P at the middle and is 

 supported at the ends, the only couples at the ends being 

 such torsional couples as will keep the depth vertical 

 there. 



To make the conditions precise it should be stated that the 

 load at the middle and the supporting forces at the ends are 

 applied at the central points of the sections. 



In this case a couple N acts at each end as shown at the 

 end A in fig. 3. Also a force ^P acts at each end to support 

 the load. The couple G does not act in this case. 



Measuring x from one end the equations for the equi- 

 librium of a portion of the beam, obtained in the same way 

 as equations (14) and (15), are 



E0g = -iPat,. ...... (26) 



K 4.= N +K^H R • • • (27) 



From these we get 



that is 



^ d 2 r d 2 y 



d?r_ P 2 , 



dx 2 ~ 4EnCK^ T 



where now 



= -m\r 2 T, (28) 



™ 4 = TO- * < 29 > 



Equation {28) differs from equation (19) only in having 

 ^P instead of P. The conditions to be satisfied in this case 

 are, however, 



r = when x = 



— - =0 when x = iZ. 



dx 2 



This latter condition makes the twist a maximum at the 

 middle, which is clearly the actual state of affairs in the most 

 stable position of the beam. 



The series for t are exactly the same as in equation (22), 

 and the condition that r = when x — makes the constant b 



