of the Principle of Least Action, 433 



shows that its maximum intensity is double of its mean inten- 

 sity. The conditions necessary for the truth of this solution 

 may perhaps be approximately realized in the case of a shaft and 

 its bearing. 



4. In the most general case of the equilibrium of elasticity, U 

 the work done is given by the equation 



2TJ =flf[i4^^ N > +N ^ +N ^ 



SX + 2/j, 

 N 1 N,+N,N, + N g N3-T 1 g -T,»-T, 



— - \dxdydz, 



where N p N g , N 3 are the intensities of the normal pressures on 

 three faces of the element perpendicular to the axes of x, y, z 

 respectively, T p T 2 , T 3 the intensities of the tangential stress on 

 the element parallel to the planes yz, xz, xy respectively, and X/u, 

 are constants depending on the nature of the material which is 

 supposed homogeneous, and of perfect elasticity. The notation, 

 to facilitate comparison, is the same as that employed by M.Lame. 

 This value of U must now be made a minimum, subject to 

 three equations of condition necessary for the equilibrium of the 

 element, namely, 



tfN 1 + rfT3 + iT 2=0 

 dx dy dz 3 



dx dy dz ' 



dx dy dz 



(Lame, Lecons swr V Elasticity, chap. 7.) 



Applying Lagrange's process, we obtain three equations, of 

 which one is 



- 



and three other equations, of which one is 



/j, dz dy 

 4>\> <fo $3 being unknown functions of on, y } z. 



