STRAINED ELASTIC SOLIDS 487 



OX'Y'. This displacement cannot be effected by any simple 

 rotation. (A rotation of the body for example round the axis 

 of OX' through two right angles would be represented by the 

 equations 



X = x', y = -y', z = -z' 



whose U determinant has the value +1.) Indeed, to produce 

 the displacement indicated we would have to conceive a con- 

 tinuous distortion of the body in which all the particles of the 

 body would have to be gradually "squeezed" towards the plane 

 OX'Y' , the body growing flatter and more "disc-like" until it is 

 squeezed to a limiting volume zero; thereupon it would begin to 

 swell again to the same size as before, but with all the particles 

 previously on the positive side of the plane OX'Y' now on the 

 negative, and vice-versa. Such a process while conceivable is 

 hardly possible physically. It should be noted that in the 

 course of such a conceptual continuous process the volume 

 would pass through the value zero; also the determinant H, 

 which is the ratio of volume dilatation, would pass through 

 decreasing small values from unity to zero, then change to 

 negative values and grow numerically (decreasing algebraically) 

 to the limiting value —1, as we indicated above. This short 

 discussion will perhaps help the reader while perusing pages 

 210, 211. 



We now revert to the short paragraph beginning near the top 

 of page 205 with the words "In the case of isotropic bodies." 

 Unless the reader is on his guard the position of this paragraph 

 in the general argument might unconsciously incline his mind 

 to the view that the subsequent discussion concerning principal 

 axes of strain is only valid for isotropic solids, and this would be 

 unfortunate. Nothing in Gibbs' own argument nor in that given 

 earlier in this article warrants such a restriction. No mat- 

 ter what the nature of the solid, any group of external forces 

 will produce a distortion and a system of stresses such that there 

 are in any element three principal axes of strain for which the 

 shearing strain-coefficients d, Ch, ee vanish, and three principal 

 axes of stress for which the stress-constituents Yz (or Zy), 

 Zx (or Xz), Xy (or Yx) vanish. If the strain is homogeneous 



