Mi: I,'. V. sol Tl I \VKLL OX Till. HKXKRAL THKOKY OF KLASTIC STABILITY. 



ration of sli^lit distortion from the equilibrium position, which can be maintained 

 ivithout the introduction of additional stress at the boundaries, if the equilibrium of 

 the second configuration is neutral. We shall consider first a stress-system which is 

 such that the principal stresses in the second configuration have the same magnitudes 

 and directions throughout the body ;* and we shall take these directions as axes of 

 x, y and z. We may then define the second and third configurations by saying that 

 in them the co-ordinates of the point (x, y, z) become 



and 



respectively. We shall not limit the values of e t , e 3 , ? 3 , although in practical cases 

 they must be small : u', i/, / are infinitesimal. In the second configuration the axes 

 Ox, Oy, Oz are directions of principal stress, and the stresses are 



2C 

 X, = gKni-lJcj + Cj + eJ, ...,&c (2.) bis 



In the third configuration we shall find that lines which in the first configuration 

 were slightly inclined to Ox, Oy, Oz become directions of principal stress and strain. 

 The final extension of a line which originally had direction-cosines /, m, n is 



It may be shown that e' has a stationary value when 



(!+,) 



m = m, = 



and 



(i+i)=! 



I/ *S 



n = n, = 



(5) 



to terms of the first order in n', v', w/.t 



In some cases, such as GRRENHIU/S problem of the stability of a heavy vertical rod (p. 188, footnote), 

 i necessary to allow for variation in one or more of the principal stresses ; the necessary alterations are 

 easily made, and as they are not required for the examples of this paper their consideration would involve 

 unnecessary complexity. 



A,l,M May 1. The approximation of these expressions is insufficient if any two of the principal 

 (i, *, s) in the second configuration are equal ; in this case additional terms must be retained in 

 denominators. The equilibrium under hydrostatic stress (,, = e, = ,) is necessarily and obviously 

 stable.] 



