Mi:. R. V. SOUTHWELL ON TIIK iKNT.i: AL THEORY OF ELASTIC STABILITY. 239 



harmonic terms, and when the load is applied one of these harmonics will be 

 developed very much more than the others, just as one constituent harmonic may be 

 developed by " resonance " in an alternating current wave of irregular shape. In the 

 ordinary strut problem this in.i^nifi.-il liannonir is sin-b that one-half wave occupies 

 the length of the strut, but in other problems, such .'is that of the tubular strut, 

 though there is always a " favourite " or " natural harmonic " which is especially 

 magnified, its relation to the dimensions may l>e more complicated.* In any case the 

 effects of practical imperfections of form might be studied, if the analytical difficulties 

 could t)e surmounted, by investigating the rate at which the amplitude of this 

 " natural harmonic " increases with the load, when its value in the initial configu- 

 ration is given ; and the results of the investigation might be shown graphically by 

 curves of distortion, similar in character to the curves of fig. 8, in which the aljscissae 

 represented the amplitude of the natural harmonic, and the ordinates represented the 

 magnitude of the applied stress-system, or " load." 



These " curves of distortion " are of considerable utility for the study 01 problems 

 in elastic stability, even though their true form can only be guessed. They help us, 

 for example, to explain, and in some degree to remedy, the serious discrepancy 

 existing between EULEK'S theory and the results of experiments on short struts. 

 The discrepancy has often been attributed to practical imperfections of form ; but it 

 should hardly be necessary to point out that practical imperfections are likely to 

 diminish rather than to increase in importance, as the dimensions of an elastic solid 

 become more nearly comparable, so that they will never be more effective as causes of 

 weakness than in struts of great length, which, as a matter of fact, give results in 

 close agreement with EULER'S formula. 



A more satisfactory explanation of this, and of similar discrepancies in other 

 problems, may be found in the fact that the ordinary theory of elastic stability 

 neglects the possibility of elastic break-down. If we attempt to draw " curves of 

 distortion " for any single problem, we shall find that, apart from the other data of 

 the problem, three possible cases exist, depending upon the elastic limit of the material 

 under consideration : 



(1) The material may be of infinite strength ; 



(2) Its elastic limit may be so high that the critical load, as determined by the 



theory of instability, is not sufficient to cause elastic break-down in the 

 configuration of equilibrium ; 



(3) Elastic break-down may occur, even in the position of equilibrium, at a load less 



than the critical value. 



In the first case (which is, of course, purely ideal), the distortion due to loading 

 will vanish when the loading is removed, and in this sense we may say that the 



* In the problem of the tubulur strut, the " favourite harmonic " is, of course, defined by that value of q 

 which corresponds to a minimum value of J in equations (102) or (104). 



