Mi:. U. V. sol TIIUKI.I. ON Till: OIWEEAL THEORY OF ELASTIC STABILITY. 



many ii,, r rfections which occur in practice. For simplicity, let us assume that the 

 sph.-'r.-. l"wl, ami pluiitf-r are still smooth, rigid, and accurately formed, but that 

 tli.- line of thrust of the plunger is eccentric by an amount S. It is easy to see from 

 ti- 7 that the displacement of the sphere from the line of thrust of the plunger, when 

 the system is in equilibrium under a load P, is 



d = r sin 6 = A+(R-r) sin 0,1 



where L . (110) 



_P _ tan . 



W ~~ tan 6- tan $ 



and these equations enable us to trace the steady increase of the sphere's displacement 

 as the load on the plunger is increased from a zero value. 



o Values of d. 



i-or 



Fig. 7. 



Fig. 8. 



Thus in fig. 8 curves are drawn to connect P and d, for a value 3 of the ratio R/r, 

 when the initial displacement S has the values 0, O'Olr, and O'lr respectively. At 

 the points on these curves for which P has a maximum value, " collapse " will occur, 

 since the equilibrium then becomes unstable. The locus of these points is shown in 

 the figure by a broken line, and a dot-and-dash line shows the connection between S 

 and the maximum value of P. From the latter curve it is evident that a small 

 initial inaccuracy may cause a material reduction in the " collapsing load " ; never- 

 theless the " critical load " gives a limit which will be more and more nearly attained 

 as our experimental accuracy is improved, and its investigation is by no means useless 

 for practical purposes. 



When the problem is one of elastic stability, the discussion of imperfections by 

 analytical methods will, in general, be beyond our power ; but it is clear that similar 

 remarks will apply. An " exchange of stabilities " at some " point of bifurcation "* 

 must be regarded as a purely ideal conception, and in practice there will always be a 

 steady increase of distortion as the load is increased, owing principally to practical 

 imperfections of form. A strut, for example, may be very accurately loaded, if 

 suitable methods are employed, but its centre-line will never be quite straight ; the 

 initial deflection which characterizes it may be regarded as composed of a series of 



* H. POIXCARE, ' Acta Mathomatica,' 7. (1885), p. 259. 





