302 Mr. A. G. M. Micbell on the Mastic Stability of 



If now the cantilever be supposed fixed at the point x=l, 

 so that 6 is zero at that point, there results the equation 



4.3 + 8.7.4.3 " -4rc(4n-l)....4.3 + {) 



(in which R is put for I 4 / J) , expressing the condition for the 

 possibility of equilibrium of the cantilever in the slightly 

 displaced form, or, in other words, the condition of critical 

 stability. 



The first and second roots of (6) are 



R= 16-101 . . . ., and R = 104-98 .... 



The modes of instability corresponding to both roots are 

 illustrated in the accompanying figure, which represents the 



Fig. 1. 



projection on the plane xy of the axial plane of greatest 

 rigidity. The extended edge of the plane is represented by 

 the thicker line. The figure as a whole shows the deformed 

 shape corresponding to the second root. In this case the 

 sign of 6 changes at a distance almost exactly three-eighths 

 of the total length from the fixed end, while y is of the same 

 sign throughout. The portion of the cantilever between the 

 point at which 6 is zero and the free end is unstable in the 

 mode corresponding to the first root of equation (6). In this 

 case 6 and y are each of one sign throughout. 



III. Another set of end conditions will express the case of 

 a beam supported and prevented from turning about its axis 

 at its ends and loaded with a single transverse load at its 

 middle point. 



If 21 be the length of the beam, 2N the applied force, the 

 other symbols having the same meanings as before, is to 



be made zero when x is zero, and -j- zero when x is equal to 



/ (the middle point). 



Hence in equation (5) A = and either I is zero or 



B R 2 R" 



4 + 8.5.4 ••* -4/1(4/1-3).... 5.4 +,,, '~" ' ^ 



