DEFINITION OF HOMOGENEOUS STRAIN. 17 



If any one point of a body is fixed in space, the mass can be brought 

 from its original orientation into any other orientation by simple rotation 

 about some one axis passing through the fixed point. This is a well 

 known and very fundamental theorem, one of the many which bears 

 Euler's name. 



In homogeneous strain each elementary cube of the mass is deformed 

 in the same manner as any other; each straight line in the unstrained 

 mass therefore remains a straight line after strain, being elongated or 

 deflected to the same extent as any of the lines parallel to it, and all lines 

 originally parallel remain parallel. Hence any sphere in the unstrained 

 mass becomes an ellipsoid, and all such ellipsoids are similar. 



Irrotational strain is a term applied to a change in form and dimen- 

 sions unaccdmpanied by any change in the direction of the axes of the 

 strain ellipsoid. It is manifest that any dilation and any desired ratio 

 between the axes of the strain ellipsoid can be produced without chang- 

 ing the direction of these axes. 



Hence if the changes in a homogeneously strained elastic mass are 

 regarded relatively to any one point of it, any change in the relations of 

 its parts may be considered as compounded of a rotation about a single 

 axis into the required orientation and an irrotational strain. 



There is no necessary connection between the axes of strain and the 

 axis of rotation, and the latter will not in general coincide with any of 

 the strain axes. The rotation in the general case is resoluble into three 

 partial rotations about the three strain axes. 



For the purposes of this paper, it is both necessary and sufficient to 

 examine the conditions affecting the mass in the principal sections of the 

 strain ellipsoid. This is equivalent to selecting any one such section and 

 considering the movements relatively to it. When such a selection is 

 made, the rotations of the plane itself on axes drawn in it are eliminated, 

 and only the rotation of the mass about a line perpendicular to the plane 

 of reference retains its significance. 



The first subject of discussion therefore is an ideally elastic mass with 

 one point fixed when subjected to any distortions, however great, which 

 will produce rotation about not more than one axis of the strain ellipsoid. 



DISPLA CEMENTS. 



General Conditions. — Let the center of inertia of a mass remain at rest; 

 let any other point or points of it be moved in planes parallel to the x y 

 plane without limitation, provided only that the strain shall be homo- 

 geneous, but let every plane originally parallel to that oixy remain 

 parallel to it, so that deformation parallel to o % shall consist simply of 



