Long Beams under Transverse Forces. 307 



In this case the unstable form of the beam consists of an 

 even number of loops 



If in these equations h is made very large, each of the 

 single forces becomes equivalent to a couple applied at the 

 end of the beam of magnitude 



Ph=L. 

 In this case 



If, on the other hand, h becomes vanishingly small 



1 P_ 

 e* ~ &' 



and the condition of instability in the first case becomes 



;2_ ™ 2 7r 2 /3 2 



'--4P-' 



or 



p= n 2 7r^ 2 



4/ 2 ' 

 or for the first mode of instability n is equal to 1, and 



P= 



7T 2 /3 2 



4/ 2 ' 



which is Euler's well-known result for the unstable load, P 1? 

 of a column of length 21. 



Experimental Verification. 



VII. The conditions for the instability of beams in the 

 modes discussed above probably admit of more accurate 

 realization than those for any other known instabilities of 

 elastic solids. 



I have made the attempt to verify one or two of the results 

 given by the theory. 



The test-beam experimented upon was one of Chesterman's 

 engineer's steel straight-edges, from which the feathered edge 

 was removed by planing. The tests were made on the 

 middle" portion of a bar 4 feet long in all. The tested portion 

 had a mean width of 4*3(>7 cms., and mean thickness of 

 "2591 cm. over a length of 110 cms. 



The elastic constants of the material were determined by 

 bending and twisting the specimen itself. The deflexions in 

 the bending experiments were measured by a screw-micrometer 



